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PARTIAL 

DIFFERENTIAL EQUATIONS 
OF MATHEMATICAL PHYSICS 

By H. BATEMAN 

This truly encyclopedic exposition of the 
methods of solving boundary value problems of 
mathematical physics by means of definite ana 
lytical expressions is valuable as both text and 
reference work. 

It covers an astonishingly broad field of prob- 
, lems and contains full references to classical 
and contemporary literature, as well as numerous 
examples on which the reader may test his skill. 
This edition contains a number of corrections 
and additional references furnished by the late 
Professor Bateman. 

The book includes sections on: 

Relation of the differential equations to varia 
tional principles, approximate solution of bound- 
ary value problems, method *of Ritz, orthogonal 
functions; classical equations, including uniform 
motion, Fourier series, free and forced vibrations, 
Heaviside's expansion, wave motion, potentials, 
Laplace's equation. 

Applic?^ns of the theorems of Green and 
Stokes; ' ><nann's method, elastic solids, fluid 
motion, torsion, membranes, electromagnetism; 
two dimensional problems, Fourier inversion, vi- 
bration of a loaded string and of a shaft; con- 
formal mapping, including the Riemann theorem, 
the distortion theorem, mapping of polygons. 

Equations in three variables, wave motion, 
teat flow; polar co-ordinates, Legendre polyno- 
nials, with applications; cylindrical co-ordinates, 
diffusion, vibration of a circular membrane; el- 
liptic and parabolic co-ordinates, with the cor- 
responding boundary problems; torodial co-ordi- 
nates and applications, 

". . the book must be in the hands of every 
one who is interested in the boundary value 
problems of mathematical physics'*. Bulletin 
of American Mathematical Society. 

Text in English. 6x9. xxii-f-522 pages, 29 
illustrations. Originally published at $10.50. 



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PARTIAL 
DIFFERENTIAL EQUATIONS 

OF 

MATHEMATICAL PHYSICS 



BY 

H. BAT EM AN, M.A., PH.D. 

Late Fellow of Trinity College, Cambridge ; 

Professor of Mathematics, Theoretical Physics 

and Aeronautics, California Institute of Technology, 

Pasadena, California 



NEW YORK 
DOVER PUBLICATIONS 

1944 



First Edition 1932 



First American Edition 1944 

By special arrangement with the 
Cambridge University Press and The Macmillan Co. 



Printed in the U. S. A. 



Dedicated 

to 
MY MOTHER 



CONTENTS 

PREFACE page xiii 

INTRODUCTION xv-xxii 

CHAPTER I 

THE CLASSICAL EQUATIONS 

1-11-1-14. Uniform motion, boundary conditions, problems, a passage to the 

limit. 1-7 

1-15-1-19. Fourier's theorem, Fourier constants, Cesaro's method of summation, 
Parseval's theorem, Fourier series, the expansion of the integral of a bounded 
function which is continuous bit by bit. . 7-16 

1-21-1-25. The bending of a beam, the Green's function, the equation of three 
moments, stability of a strut, end conditions, examples. 16-25 

1*31-1-36. F^ee undamped vibrations, simple periodic motion, simultaneous 
linear equations, the Lagrangian equations of motion, normal vibrations, com- 
pound pendulum, quadratic forms, Hermit ian forms, examples. 25-40 

1-41-1 -42. Forced oscillations, residual oscillation, examples. 40-44 

1-43. Motion with a resistance proportional to the velocity, reduction to alge- 
braic equations. 44 d7 

1-44. The equation of damped vibrations, instrumental records. 47-52 

1-45-1 -46. The dissipation function, reciprocal relations. 52-54 

1-47-1-49. Fundamental equations of electric circuit theory, Cauchy's method 

of solving a linear equation, Heaviside's expansion. 54-6Q 

1-51 1-56. The simple wave-equation, wave propagation, associated equations, 
transmission of vibrations, vibration of a building, vibration of a string, torsional 
oscillations of a rod, plane waves of sound, waves in a canal, examples. 60-73 

1-61-1 -63. Conjugate functions and systems of partial differential equations, 
the telegraphic equation, partial difference equations, simultaneous equations 
involving high derivatives, examplu. 73-77 

1-71-1-72. Potentials and stream-functions, motion of a fluid, sources and 
vortices, two-dimensional stresses, geometrical properties of equipotentials and 
lines of force, method of inversion, examples. 77-90 

1-81-1-82. The classical partial differential equations for Euclidean space, 
Laplace's equation, systems of partial differential equations of the first order 
fchich lead to the classical equations, elastic equilibrium, equations leading to the 
^uations of wave-motion, 90-95 

S 1*91. Primary solutions, Jacobi's theorem, examples. 95-100 

$1'92./ The partial differential equation of the characteristics, bicharacteristics 

and rays. 101-105 

; 1 '93-1 94. Primary solutions of the second grade, primitive solutions of the 
wave-equation, primitive solutions of Laplace's equation. 105-111 

1-95. Fundamental solutions, examples. 111-114 



viii Contents 

CHAPTER n 

APPLICATIONS OF THE INTEGRAL THEOREMS OF GREEN AND STOKES 

2*11-2-12. Green's theorem, Stokes' s theorem, curl of a vector, velocity 
potentials, equation of continuity. pages 116-118 

2-13-2-16. The equation of the conduction of heat, diffusion, the drying of 

wood, the heating of a porous body by a warm fluid, Laplace's method, example. 118-125 

2-21-2*22. Riemann's method, modified equation of diffusion, Green's func- 
tions, examples. 126-131 

<f 2-23-2*26. Green' s^theorem^for a general linear differential equation of the 
second order, characteristics, classification of partial differential equations of the 
second order, a property of equations of elliptic type, maxima and minima of 
solutions. 131-138 

2-31-2-32. Green's theorem for Laplace's equation, Green's functions, reciprocal 
relations. ~ " 138-144 

2-33-2-34. Partial difference equations, associated quadratic form, the limiting 

process, inequalities, properties of the limit function. 144-152 

2-41-2-42. The derivation of physical equations from a variational principle, 
Du Bois-Reymond's lemma, a fundamental lemma, the general Eulerian rule, 
examples. 152-157 

2-431-2-432. The transformation of physical equations, transformation of 
Eulerian equations, transformation of Laplace's equation, some special trans- 
formations, examples. 157-162 

2-51. The equations for the equilibrium of an isotropic elastic solid. 162-164 

2-52. The equations of motion of an inviscid fluid. 164-166 

2-53. The equations of vortex motion and Liouville's equation. . 166-169 

2-54. The equilibrium of a soap film, examples. 169-171 

2-56-2-56. The torsion of a prism, rectilinear viscous flow, examples. 172-176 

2-57. The vibration of a membrane. 176-177 

2-58-2-59. The electromagnetic equations, the conservation of energy and 
momentum in an electromagnetic field, examples. 177-183 

2-61-2-62. Kirchhoflfs formula, Poisson's formula, examples. 184-189 

2-63-2-64. Helmholtz's formula, Volterra's method, examples. 189-192 

2-71-2-72. Integral equations of electromagnetism, boundary conditions, the 
retarded potentials of electromagnetic theory, moving electric pole, moving electric 
and magnetic dipoles, example. 192-201 

2-73. The reciprocal theorem of wireless telegraphy. 201-203 



Contents 



IX 



CHAPTER IH 
TWO-DIMENSIONAL PROBLEMS 

3*11. Simple solutions and methods of generalisation of solutions, example, pages 204-207 
3*12. Fourier's inversion formula. 207-211 

3-13-3-15. Method of summation, cooling of fins, use of simple solutions of a 
complex type, transmission of vibrations through a viscous fluid, fluctuating tem- 
peratures and their transmission through the atmosphere, examples. 211-215 
3-16. Poisson's identity, examples. 215-218 
3*17. Conduction of heat in a moving medium, examples. 218221 
3-18. Theory of the unloaded cable, roots of a transcendental equation, Kosh- 
liakov's theorem, effect of viscosity on sound waves in a narrow tube. 221-228 
3-21. Vibration of a light string loaded at equal intervals, group velocity, 
electrical filter, torsional vibrations of a shaft, examples. 228-236 
3-31-3-32. Potential function with assigned values on a circle, elementary 
treatment of Poisson's integral, examples. 236-242 
3-33-3-34. Fourier series which are conjugate, Fatou's theorem, Abel's theorem 
for power series. 242-245 
3-41. The analytical character of a regular logarithmic potential. 245-246 
3-42. Harnack's theorem. 246-247 
3-51. Schwarz's alternating process. 247-249 
3-61. Flow round a circular cylinder, examples. 249-254 
3-71. Elliptic coordinates, induced charge density, Munk's theory of thin 
aerofoils. 254r~260 
3-81-3-83. Bipolar co-ordinates, effect of a mound or ditch on the electric 
potential, example, the effect of a vertical wall on the electric potential. 260-265 



CHAPTER IV 
CONFORMAL REPRESENTATION 

4-1 1-4-21. Properties of the mapping function, invariants, Riemann surfaces 
and winding points, examples. 266-270 

4-224-24. The bilinear transformation, Poisson's formula and the mean value 
theorem, the conformal representation of a circle on a half plane, examples. 270-275 

4-31-4-33. Riemann' s problem, properties of regions, types of curves, special 
and exceptional cases of the problem. 275-280 

4-41-4-42. The mapping of a unit circle on itself, normalisation of the mapping 
problem, examples. 280-283 

4-43. The derivative of a normalised mapping function, the distortion theorem 
and other inequalities. 283-285 

4-44. The mapping of a doubly carpeted circle with one interior branch point. 285-287 
4-45. The selection theorem. 287-291 

4-46. Mapping of an open region. 291-292 

b-2 



x Contents 

4-51. The Green's function. pages 292-294 

4-61-4-63. Schwarz's lemma, the mapping function for a polygon, mapping of 

a triangle, correction for a condenser, mapping of a rectangle, example. 294-305 

4-64. Conformal mapping of the region outside a polygon, example. 305-309 

4-71-4-73. Applications of conformal representation in hydrodynamics, the 
mapping of a wing profile on a nearly circular curve, aerofoil of small thickness. 309-316 

4-81-4-82. Orthogonal polynomials connected with a given closed curve, the 
mapping of the region outside C', examples. 316-322 

4-91-4-93. Approximation t*> the mapping function by means of polynomials, 
Daniell's orthogonal polynomials, Fe JOT'S theorem. 322-328 

CHAPTER V 
EQUATIONS IN THREE VARIABLES 

5-11-5-12. Simple solutions and their generalisation, progressive waves, 
standing waves, example. 329-331 

5-13. Reflection and refraction of electromagnetic waves, reflection and refrac- 
tion of plane waves of sound, absorption of sound, examples. 331-338 

5-21. Some problems in the conduction of heat. 338-345 

5-31. Two-dimensional motion of a viscous fluid, examples. 345-350 



CHAPTER VI 

POLAR CO-ORDINATES 

6-11-6-13. The elementary solutions, cooling of a solid sphere. 351-354 

6-21-6-29. Legendre functions, Hobson's theorem, potential function of degree 
zero, Hobson's formulae for Legendre functions, upper and lower bounds for the 
function P n (/-i), expressions for the Legendre functions as nth derivatives, the 
associated Legendre functions, extensions of the formulae of Rodrigues and 
Conway, integral' relations, properties of the Legendre coefficients, examples. 354-366 

6-316-36. Potential function with assigned values on a spherical surface $, 
derivation of Poisson's formula from Gauss's mean value theorem, some applica- 
tions of Gauss's mean value theorem, the expansion of a potential function in a 
series of spherical harmonics, Legendre's expansion, expansion of a polynomial in 
a series of surface harmonics. 367-375 

6-41-6-44. Legendre functions and associated functions, definitions of Hobson 
and Barnes, expressions in terms of the hypergeometric function, relations be- 
tween the different functions, reciprocal relations, potential functions of degree 
n + \ where n is an integer, conical harmonics, Mehler's functions, examples. 375-384 

6-51-6-54. Solutions of the wave-equation, Laplace's equation in n + 2 variables, 
extension of the idea of solid angle, diverging waves, Hankel's cylindrical 
functions, the method of Stieltjes, Jacobi's polynomial expansions, Wangerin's 
formulae, examples. 384-395 

6-61. Definite integrals for the Legendre functions. 395-397 



Contents xi 



CHAPTER VII 
CYLINDRICAL CO-ORDINATES 

7*11. The diffusion equation in two dimensions, diffusion from a cylindrical rod, 
examples. pages 398-399 

7-12. Motion of an incompressible viscous fluid in an infinite right circular 
cylinder rotating about its axis, vibrations of a disc surrounded by viscous fluid, 
examples. 399-401 

7 13. Vibration of a circular membrane. 401 

7-21. The simple solutions of the wave-equation, properties of the Bessel 
functions, examples. 402-404 

7-22. Potential of a linear distribution of sources, examples. 404-405 

7-31. Laplace's expression for a potential function which is symmetrical about an 
axis and finite on the axis, special cases of Laplace's formula, extension of the 
formula, examples. 405-409 

7-32. The use of definite integrals involving Bessei functions, Sommerf eld's 
expression for a fundamental wave-function, Hankel's inversion formula, 
examples. 409-412 

7-33. Neumann's formula, Green's function for the space between two parallel 

planes, examples. * 412-415 

7-41. Potential of a thin circular ring, examples. 416-417 

7*42. The mean value of a potential function round a circle. 418-419 

7-51. An equation which changes from the elliptic to the hyperbolic type. 419-420 

CHAPTER VIII 
ELLIPSOIDAL CO-ORDINATES 

8-1 1-8-12. Confocal co-ordinates, special potentials, potential of a homogeneous 

solid ellipsoid, Maclaurin's theorem. 421-425 

8-21 . Potential of a solid hypersphere whose density is a function of the distance 

from the 6entre. 426-427 

8-31-8-34. Potential of a homoeoid and of an ellipsoidal conductor, potential 
of a homogeneous elliptic cylinder, elliptic co-ordinates, Mathieu functions, 
examples. 427-433 

8-41-8-45. Prolate spheroid, thin rod, oblate spheroid, circular disc, con- 
ducting ellipsoidal column projecting above a flat conducting plane, point charge 
above a hemispherical boss, point charge in front of a plane conductor with a pit 
or projection facing the charge. 433-439 

8-51-8-54. Laplace's equation in spheroidal co-ordinates, Lame products for 
spheroidal co-ordinates, expressions for the associated Legendre functions, 
spheroidal wave-functions, use of continued fractions and of integral equations, 
a relation between spheroidal harmonics of different types, potential of a disc, 
examples. 440-448 



Xll 



Contents 



CHAPTER IX 

PARABOLOIDAL CO-ORDINATES 

9-11. Transformation of the wave-equation, Lam6 products. pages 449-451 

0-21-0-22. Sonine's polynomials, recurrence relations, roots, orthogonal pro- 
perties, Hermite's polynomial, examples. 451-455 
0-31. An erpression for the product of two Sonine polynomials, confluent 
hypergeometric functions, definite integrals, examples. 455-460 

CHAPTER X 

TOROIDAL COORDINATES 

10-1 . Laplace's equation in toroidal co-ordinates, elementary solutions, examples. 461-462 
10-2. Jacobi's transformation, expressions for the Legendre functions, examples. 463-465 
10-3. Green's functions for the circular disc and spherical bowl. 465-468 
10-4. Relation between toroidal and spheroidal co-ordinates. 468 
10-5. Spherical lens, use of the method of images, stream-function. 468-472 
10-6-10-7. The Green's function for a wedge, the Green's function for a semi- 
infinite plane. 472-474 
10-8. Circular disc in any field of force. 474-475 



CHAPTER XI 
DIFFRACTION PROBLEMS 

11-1-11-3. Diffraction by a half plane, solutions of the wave-equation, Som- 

merfeld's integrals, waves from a line source, Macdonald's solution, waves from 

a moving source. 

11-4. Discussion of Sommerf eld's solution. 

11 -5-1 1-7. Use of parabolic co-ordinates, elliptic co-ordinates. 



476-483 
483-486 
486-490 



CHAPTER XII 
NON-LINEAR EQUATIONS 

12*1. Riccati's equation, motion of a resisting medium, fall of an aeroplane, 
bimolecular chemical reactions, lines of force of a moving electric pole, examples. 491-496 
12-2. Treatment of non-linear equations by a method of successive approxi- 
mations, combination tones, solid friction. 497-501 
12*3. The equation for a minimal surface, Plateau's problem, Schwarz's 
method, helicoid and catenoid. 501-509 
12-4. The steady two-dimensional motion of a compressible fluid, examples. 509-511 



APPENDIX 

LIST OF AUTHORS CITED 

INDEX 



512-514 
515-519 
520-522 



PREFACE TO AMERICAN EDITION 

The publishers are gratified that through the cooperation of The 
Macmillan Company and the Cambridge University Press, Professor 
Bateman's definitive work on partial differential equations is made 
available for text and reference use by American mathematicians and 
physicists in a reduced price, corrected edition. 

Professor Bateman t has kindly provided the corrections and addi- 
tions to references which are found on pages 11, 15, 24, 49, 50, 73, 
124, 126, 127, 212, 219, 229, 230, 247, 261, 359, 364, 366, 403, 404, 
410, 415, 417, 452, 454, 457, 462, 464, 477, 515, 517, 518 and 519. 
The correction required on page 219 was brought to Professor 
Bateman's attention by Mr. W. H. Jurney. 

DOVER PUBLICATIONS 



PREFACE 

IN this book the analysis has been developed chiefly with the aim of 
obtaining exact analytical expressions for the solution of the boundary 
problems of mathematical physics. 

In many cases, however, this is impracticable, and in recent years much 
attention has been devoted to methods of approximation. Since these are 
not described in the text with the fullness which they now deserve, a brief 
introduction has been written in which some of these methods are sketched 
and indications are given of portions of the text which will be particularly 
useful to a student who is preparing to use these methods. 

No discussion has been given of the partial differential equations which 
occur in the new quantum theory of radiation because these have been well 
treated in several recent book^, and an adequate discussion in a book of 
this type would have greatly increased its size. It is thought, however, that 
some of the analysis may prove useful to students of the new quantum 
theory. 

Some abbreviations jmd slight departures from the notation used in 
recent books have been adopted. Since the /,-notation for the generalised 
Laguerre polynomial has been used recently by different writers with 
slightly different meanings, the original T-notation of Sonine has been 
retained as in the author's Electrical find Optical Wave Motion. It is 
thought, however, that a standardised //-notation will eventually be 
adopted by most writers in honour of the work of Lagrange and Laguerre. 

The abbreviations '"eit" and "eif " used in the text might be used with 
advantage in the new quantum theory, together with some other abbrevia- 
tions, such as "oil" for eigenlrsrei and k 'eiv" for eigenvector. 

The Heaviside Calculus and Ihe theory of integral equations are only 
briefly mentioned in the text: they belong rather to a separate subject 
which might be called the Integral Equations of Mathematical Physics. 
Accounts of the existence theorems of potential theory, Sturm-Liouville 
expansions and ellipsoidal harmonics have also been omitted. Many 
excellent books have however appeared recently in which these subjects 
are adequately treated. 

I feel indeed grateful to the Cambridge University Press for their very 
accurate work and intelligent assistance during the printing of this book. 

M. BATEMAN 
November 1931 



INTRODUCTION 

THE differential equations of mathematical physics are now so numerous 
and varied in character that it is advisable to make a choice of equations 
when attempting a discussion. 

The equations considered in this book are, I believe, all included in some 
set of the form x x ~ <> F 

- - - - 

to ' .' " to. ' 

where the quantities on the left-hand sides of these equations are the 
variational derivatives* of a quantity F 9 which is a function of I independent 
variables x lt ... x lt of m dependent variables y l9 "... y m and of the deriva- 
tives up to order n of the t/'s with respect to the x's. The meaning of a 
variational derivative will be gradually explained. 

(T) The first property to be noted is that the variational derivatives of a 
function F all vanish identically when the function can be expressed in the 

form F _do l dG t da t 

JL j r ~j r i j j 
dx-i cLx^ axi 

where each of the functions G 8 is a function of the x's and t/'s and of the 
derivatives up to order n 1 of the i/'s with respect to the x'a. The notation 
d/dx 8 is used here for a complete differentiation with respect to x 8 when 
consideration is taken of the fact that F is not only an explicit function of 
x s but also an explicit function of quantities which are themselves functions 
of x 8 . 

Another statement of the property just mentioned is that the varia- 
tional derivatives of F vanish identically when the expression 

F dx l dx 2 ... dx l 
is an exact differential. 

In the case when there is only one independent variable x and only one 
dependent variable y, whose derivatives up to order n are respectively 
y'> y"y y (n) 9 the condition that Fdx may be an exact differential is 
readily found to be 



- 
dy dx \dy' dx*\dy" 

Now the quantity on the left-hand side of this equation is indeed the 
variational derivative of F with respect to y and will be denoted by the 
symbol SF/Sy. 

* For a systematic discussion of variational derivatives reference may be made to the papers 
of Th. de Donder in Bulletin de VAccuttmie BoyaU de Bdgique, Claase des Sciences (5), t. xv 
(1929-30). In some cases a set of equations must be supplemented by another to give all the 
equations in a set of the variational form. 



xvi Introduction 

In the case when F is of the form 

yN. (z) - zM x (y), 

where z is a function of x and M x (y), N x (z) are linear differential expres- 
sions involving derivatives up to order n and coefficients of these deriva- 
tives which are functions of x with a suitable number of continuous 
derivatives, we can say that the differential expressions M x (y), N x (z) are 
adjoint when 8F/8y ~ for all forms of the function z. When z is chosen 
to be a solution of the differential equation N x (z) = the expression 
zM x (y) dx is an exact differential and so z is an integrating factor of the 
differential equation adjoint to N x (z) = 0. The relation between two 
adjoint differential equations is, moreover, a reciprocal one, 

The idea of adjoint differential expressions was introduced by Lagrange 
and extended by Riemann to the case when there is more than one inde- 
pendent variable. Further extensions have been made by various writers 
for the case when there are several dependent variables*. Adjoint 
differential expressions and adjoint differential equations are now of great 
importance in mathematical analysis. 

A second important property of the variational derivatives may be 
introduced by first considering a simple integral 



and its first variation 



Integrating the (s-f l)th term s times by parts, making use of the 
equations 



it is readily seen that the portion of 87 which still remains under the sign of 
integration is 



It is readily understood now why the name " variational derivative " is 
used. The variational derivative is of fundamental importance in the 
Calculus of Variations because the Eulerian differential equation for a 
variational problem involving an integral of the above form is obtained by 
equating the variational derivative to zero. 

This rule is capable of extension, and rules for writing down the 
variational derivatives of a function F in the general case when there are 

* See for instance J. Kiirechak, Math. Ann. Bd. Lxn, S. 148 (1906); D. R. Davis, Trans. 
Amer. Math. Soc. vol. xxx, p. 710 (1928). 



Introduction xvii 

I independent variables and m dependent variables can be derived at once 
from the rules of 2-42 for the derivation of the Eulerian equations. 

Since our differential equations are always associated with variational 
problems, direct methods of solving these problems are of great interest. 

The important method of approximation invented by Lord Rayleigh* 
and developed by W. Tlitzf is only briefly mentioned in the text, though it 
has been used by RitzJ, Timoshenko and many other writers|! to obtain 
approximate solutions of many important problems. An adequate dis- 
cussion by means of convergence theorems is rather long and difficult, and 
has been omitted from the text largely for this reason and partly because 
important modifications of the method have recently been suggested which 
lead more rapidly to the goal and furnish means of estimating the error of 
an approximation. 

In Ritz's method a boundary problem for a differential equation 
D (u) = is replaced by a variation problem in which a certain integral / 
is to be made a minimum, the unknown function u being subject to certain 
supplementary conditions which are usually linear boundary conditions 
and conditions of continuity. The function u a , used by Ritz as an approxi- 
mation for u, is not generally a solution of the differential equation, but it 
does satisfy the boundary conditions for all values of the arbitrary con- 
stants which it contains. The result is that when an integral I a is calculated 
from u a in the way that / is to be calculated from u, the integral I a is 
greater than the minimum value I m of /, even when the arbitrary con- 
stants in u a are chosen so as to make I a as small as possible. This means 
that I m is approached from above by integrals of the type I a . 

Now it was pointed out by R. Courant ^[ that I m can often be approached 
from below by integrals I b calculated from approximation functions u b 
which satisfy the differential equation but are subject to less restrictive 
supplementary conditions. If, for instance, u is required to be zero on the 
boundary, the boundary condition may be loosened by merely requiring 
u b to give a zero integral over the boundary in each of the cases in which it 
is first multiplied by a function v s belonging to a certain finite set. This idea 
has been developed by Trefftz** who uses the arithmetical mean of I a and I b 

* Phil. Trans. A, vol. CLXI, p. 77 (1870); Scientific Papers, vol. I, p. 57. 

t W. Ritz, Crette, Bd. cxxxv, S. 1 (1908); (Euvres, pp. 192-316 (Gauthier-Villars, Paris, 1911). 

J W. Ritz, Ann. der Phys. (4), Bd. xxvm, S. 737 (1909). 

8. Timoshcnko, Phil. Mag. (6), vol. XLVII, p. 1093 (1924); Proc. London Math. Soc. (2), 
vol. XX, p. 398 (1921); Trans. Amcr. Soc. Civil Engineers, vol. LXXXVII, p. 1247 (1924); Vibration 
Problems in Engineering (D. Van Nostrand, New York, 1928). 

|| See especially M. Plancherel, Bull, dcs Sciences Math. t. XLVII, pp. 376, 397 (1923), t. XLVIII, 
pp. 12, 58, 93 (1924); Comptcs JRcndus, t. CLXIX, p. 1152 (1919); R. Courant, Acia Math. t. XLIX, 
p. 1 (1926); K. Friedrichs, Math. Ann. Bd. xcvni, S. 205 (1927-8). 

U R. Courant, Math. Ann. Bd. xcvn, S. 711 (1927). 

** E. Trefftz, Int. Congress o Applied Mechanics, Zurich (1926), p. 135; Math. Ann. Bd. c, 
S. 603 (1928). 



xviii Introduction 

as a close approximation for I m , and uses the difference of I a and I b as an 
upper bound for the error in this method of approximation. This method is 
simplified by a choice of functions v 8 which will make it possible to find 
simple solutions of the differential equation for the loosened boundary 
conditions. Sometimes it is not the boundary conditions but the conditions 
of continuity which should be loosened, and this makes it advisable not to 
lose interest in a simple solution of a differential equation because it does 
not satisfy the requirements of continuity suggested by physical con- 
ditions. 

In order that Ritz's method may be used we must have a sequence of 
functions which satisfy the boundary conditions and conditions of con- 
tinuity peculiar to the problem in hand. It is advantageous also if these 
functions can be chosen so that they form an orthogonal set, To explain 
what is meant by this we consider for simplicity the case of a single 
independent variable x. The functions 0! (#), ^ 2 (#)> > defined in an 
interval a < x < 6, are then said to form a normalised orthogonal set when 
the orthogonal relations 



= 1, ra= n 

are satisfied for each pair of functions of the set. This definition is readily 
extended to the case of several independent variables and functions 
defined in a domain R of these variables; the only difference is that the 
simple integrals are replaced by integrals over the domain of definition. The 
definition may be extended also to complex functions i/r n (x) of the form 
a n ( x ) -f ifin (#)> where a n (x) and /? n (x) are real. The orthogonal relations 
are then of type 



s f 

Ja 



(x) + ip m (x)] [a n (x) - ip n (x)] dx=0, m*n, 

= 1, m = n. 

Many types of orthogonal functions are studied in this book. The 
trigonometrical functions sin (nx) 9 cos (nx) with suitable factors form an 
orthogonal set for the interval (0, 2?r), the Legendre functions P n (x), with 
suitable normalising factors, form an orthogonal set for the interval 
( 1, 1), while in Chapter ix sets of functions are obtained .which are 
orthogonal in an infinite interval. The functions of Laplace, which form the 
complete system of spherical harmonics considered in Chapter vi, give an 
orthogonal set of functions for the surface of a sphere of unit radius, and it 
is easy to construct functions which are orthogonal in the whole of space. 
In Chapter iv methods are explained by which sets of normalised orthogonal 
functions may be associated with a given curve or with a given area. In 
many cases functions suitable for use in Ritz's method of approximation 



Introduction xix 

are furnished by the Lam6 products defined in Chapters m-xi. These pro- 
ducts are important, then, for both the exact and the approximate solution 
of problems. It was shown by Ritz, moreover, that sometimes the functions 
occurring in the exact solution of one problem may be used in the ap- 
proximate solution of another; the functions giving the deflection of a 
clamped bar were in fact used in the form of products to represent the 
approximate deflection of a clamped rectangular plate. 

Early writers* using Ritz's method were content to indicate the degree 
of approximation obtainable by applying the method to problems which 
could be solved exactly and comparing the approximate solution with the 
exact solution. This plan is somewhat unsatisfactory because the examples 
chosen may happen to be particularly favourable ones. Attempts have, 
however, been made by Krylofff and others to estimate the error when an 
approximating function of order n, say 

U a = $0 (%) + C l<Al ( X ) + C 2^2 (X)+ ..* + Cntn 0*0, 

is substituted in the integral to be minimised and the coefficients c s are 
chosen so as to make the resulting algebraic expression a minimum. 
Attempts have been made also to determine the order n needed to make 
the error less than a prescribed quantity e. 

,- In Ritz's method a boundary problem for a given differential equation 
must first of all be replaced by a variation problerp. There are, however, 
modifications of Ritz's method in which this step is avoided. If, for in- 
stance, the differential equation is a variational equation 8F/8u = 0, the 
same set of equations for the determination of the constants c s is obtained 
by substituting the expression U a for u directly in the equation 

tb 

Su . (SF/Su) dx = 0, 

J a 

and equating to zero the coefficients of the variations 8c s . 

This method has been recommended by HenckyJ and Goldsbrough ; it 
has the advantage of indicating a reason why in the limit the function u a 
should satisfy the differential equation. 

Another method, proposed by Boussinesq|| many years ago, has been 
called the method of least squares. If the differential equation is 

L 9 (u) = f(x), 
and a < x < b is the range in which it is to be satisfied with boundary 

* See, for instance, M. Paschoud, Sur V application de la mtihode de W. Ritz: These (Gauthier- 
Villars, Paris, 1914). 

f N. Kryloff, Comptes Rendus, t. CLXXX, p. 1316 (1925), t. CLXXXVI, p. 298 (1928); Annales 
de Toulouse (3), t. xix, p. 167 (1927). 

J H. Hencky, Zeits.fur angew. Math. u. Mech. Bd. TO, S. 80 (1927). 

G. R. Goldsbrough, Phil. Mag. (7), vol. vn, p. 333 (1929). 

|| J. Boussinesq, Ttieorie de la chaleur, 1. 1, p. 316. 



xx Introduction 

conditions at the ends, the constants c 8 in an approximating function u a , 
which satisfies these boundary conditions, are chosen so as to make the 
integral 

[L m (u a )-f(x)]*dx 

J a 

as small as possible. The accuracy of this method has been studied by 
Kryloff* who believes that Ritz's method and the method of least squares 
are quite comparable in usefulness. The method of least squares is, of 
course, closely allied to the well-known method of approximating to a 
function / (x) by a finite series of orthogonal functions 



the coefficients c s being chosen so that the integral")* 

rb 

If (*) - Ci<Ai 0*0 - C 2 </r 2 (X) - ... ~ C n ifj n (X)] 2 dx 



may be as small as possible. The conditions for a minimum lead to the 
equations & 

'.= f(s)j.(x)d8 (=l,2,...n). 

Ja 

For an account of such methods of approximation reference may be 
made to recent books by Dunham Jackson J, S. Bernstein and dc la Vallee 
Poussin||. 

In the discussion of the convergence of methods of approximation 
there is an inequality due to Bouniakovsky and Schwarz which is of 
fundamental importance. If the functions f (x), g (x) and the parameter c 
are all real, the integral 

f [/ (x) + ^ (#)] 2 = A -f 2cH + c 2 B 

Ja 

is never negative and so AB H 2 > 0. This gives the inequality 
f [/ (x)Y dx ' b \jy (x)Y dx > f [/ (x) g (x)Y dx. 

J a * a J a 

There is a similar inequality for two complex functions f(x),g (x) and 

* N. Kryloff, Comptes Rend as, t. CLXI, p. 558 (1915); t. CLXXXI, p. 86 (11)25). Sec also Krawt- 
chouk, ibid. t. c'LXXxm, pp. 474, 992 (1926). 

t G. I'larr, Comptes Rcttilus, t. XLIV, p. 984 (1857); A* Tocpler, Arizctycr dcr Kais. Akad. zu 
Wu'n (1870), p. 205. 

| 1). Jackson, "The Theory of Approximation," Arner. Math. tioc. Colloquium publications, 
vol. xi (1930). 

S. Hernstt'in, Lv^ona sur le# proprn'te* extremal?* rt In mcilleure approximation dcs fotictions 
unalytiqiu's <T une van able reelle (Gauthier-Villars, Paris, 1926). 

|| C. J . do la Vallee Poussm, Lemons sur I' approximation des foncttons d'une variable reelle (ibid. 
1919). 



Introduction xxi 

their conjugates / (x), g (x). Indeed, if c and c are conjugate complex 
quantities, the integral 



is never negative. Writing 

f=l+im, g=p+iq, c=+iri, 
where I, m, p, q, , 77 are all real, the integral may be written in the form 

A ( 2 + r? 2 ) + 2B-f 2^ + D= ~[(A + )*+ (A*i + < 
where A = (p 2 + q 2 ) dx = gg dx, 

J a Ja 

D = f (i 2 + m 2 ) dx = ( //da;, 

' Ja J a 

B + iC = f fgdx, B - iC = [ 

Ja Ja 

Since the integral is never negative, we have the inequality 

AD > B* + C 2 , 
which may be written in the form* 



In this inequality the functions / and g may be regarded as arbitrary 
integrable functions. This inequality and the analogous inequality for 
finite sums are used in 4-81. 

In the approximate treatment of problems in vibration the natural 
frequencies are often computed with the aid of isoperimetric variation 
problems. Ritz's method is now particularly useful. If, for instance, the 
differential equation is 



and the end conditions u (a) = 0, u (b) = 0, 
the aim is to make the integral 



a minimum when the integral 

f [(*)] fe 

Ja 

* This inequality is called Schwarz's inequality by E. Schmidt, Rend. Palermo, t. xxv, p. 58 
(1908). 



xxii Introduction 

has an assigned value. This is accomplished by replacing u by a finite series 
in both integrals and reducing the problem to an algebraic problem. It was 
noted by Rayleigh that very often a single term in the series will give a 
good approximation to the frequency of the fundamental frequency of 
vibration. To obtain approximate values of the frequencies of overtones it 
is necessary, however, to use a series of several terms and then the work 
becomes laborious as it is necessary to solve an algebraic equation of high 
order. Many other methods of approximating to the frequencies of over- 
tones are now available. 

Trefftz has recently introduced a new method of approximating to the 
solution of a differential equation, in which the original variation problem 
87 = is replaced by a modified variation problem 87 (<r) = in such a way 
that the desired solution u can be expressed in the form 



This method, combinfed with Trefftz's method of estimating the error of 
approximation to an integral such as / (e), can lead to an estimate of the 
error involved in a computation of u. In the problem of the deflection of 
a clamped plate under a given distribution of load, the function / (e) 
represents the potential energy when a concentrated load is placed at the 
point where the deflection u is required. 

Courant has shown that the rapidity of convergence of a method of 
approximation can often be improved by modifying the variational 
problem, introducing higher derivatives in such a way that the Eulerian 
equatioA of the problem is satisfied whenever the original differential 
equation is satisfied. This device is useful also in applications of Trefftz's 
method. 

An entirely different method of approximation is based on the use of 
difference equations which in the limit reduce to the differential equation 
of a problem. The early writers were content to adopt the principle, usually 
called Rayleigh's principle, that it is immaterial whether the limiting pro- 
cess is applied to the difference equations or their solutions. Some attempts 
have been made recently to justify this principle* and also to justify the 
use of a similar principle in the treatment of problems of the Calculus of 
Variations by a direct method, due to Euler, in which an integral is replaced 
by a finite sum. An example indicating the use of partial difference 
equations and finite sums is discussed in 2-33. 

* See the paper by N. Bogoliouboff and N. Kryloff, Annals of Math. vol. xxrx, p. 255 (1928). 
Many references to the literature are contained in this paper. In particular the method is discussed 
by R. B. Robbing, Amer. Journ. vol. xxxvn, p. 367 (1915). 



CHAPTER I 
THE CLASSICAL EQUATIONS 

1-11. Uniform motion. It seems natural to commence a study of the 
differential equations of mathematical physics with a discussion of the 
equation ^ 

3*"' 

which is the equation governing the motion of a particle which moves along 
a straight line with uniform velocity. It may be thought at first that this 
equation needs no discussion because the general solution is simply 

x = At + B, 

where A and B are arbitrary constants, but in mathematical physics a 
differential equation is almost always associated with certain supple- 
mentary conditions, and it is this association which presents the most 
interesting problems. 

A similar differential equation 



describes an essential property of a straight line, when x and y are inter- 
preted as rectangular co-ordinates, and its solution 

y = mx + c 

is the f amiliar equation of a straight line : the property in question is that 
the line has a constant direction, the direction or slope of the line being 
specified by the constant m. For some purposes it is convenient to regard 
the line as -a ray of light, especially as the conditions for the reflection and 
refraction of rays of light introduce interesting supplementary or boundary 
conditions, and there is the associated problem of geometrical foci of a 
system of lenses or reflecting surfaces. 

If a ray starts from a point Q on the axis of the system and is reflected 
or refracted at the different surfaces of the optical system it will, after 
completely traversing the system, be transformed into a second ray which 
meets the axis of the system in a point Q, which is called the geometrical 
focus of Q. The problem is to find the condition that a given point Q may 
be the geometrical focus of another given point Q. 

This problem is generally treated by an approximate < method which 
illustrates very clearly the mathematical advantages gained by means of 
simplifying assumptions. It is assumed that the angle between the ray 
and the axis is at all times small, so that it can be represented at any time 
by dy/dx. 



2 The Classical Equations 

Let y 2 '= 4a# give an approximate representation of a refracting surface 
in the immediate neighbourhood of the point (0, 0) on the axis. If y is a 
small quantity of the first order the value of x given by this equation can 
be regarded as a small quantity of the second order if a is of order unity. 
Neglecting quantities of the second order we may regard x as zero and may 
denote the slope of the normal at (#, y) by 

_ dx y 

dy 2a' 

Now let suffixes 1 and 2 refer to quantities relating to the two sides of 
the refracting surface. Since the angle between a ray and the normal to 
the refracting surface is approximately dy/dx -f- y/2a, the law of refraction 
is represented by the equations 



Denoting by [u] the discontinuity u 2 u in a quantity u, we have the 
boundary conditions 

[dy\ 



[y] = o. 

Dropping the suffixes we see that these boundary conditions are of type 



[y] = o, 

where A and B are constants which may be either positive or negative. 

In the case of a moving particle, which for the moment we shall regard 
as a billiard ball, a supplementary condition is needed when the ball strikes 
another ball, which for simplicity is supposed to be moving along the same 
line. If Ui , u 2 are the velocities of the first ball before and after collision, 
U 19 U 2 those of the second ball before and after collision, the laws of 
impact give u >- U,= - e (u, - 17,), 



mu 2 + MU 2 = 

where e is the coefficient of restitution and m, M are the masses of the two 
balls. Regarding U l as known and eliminating U 2 we have 

(M -f- m) u 2 = (m - eM) u + M (1 + e) U l . 

Replacing u 2 u by [dx/dt] we have the boundary conditions for the 
collision 

[x] =0, ( M 4- m) [g] = M ( 1 + e) ( U, - 



Boundary Conditions 3 

These hold for .the place x = x l where the collision occurs, x being the co- 
ordinate of the centre of the colliding ball. 

The boundary conditions considered so far may be included in the 
general conditions , 

' 



[y] = o, 

where A, JB and C are constants associated with the particular boundary 
under consideration. 

1-12. Other types of boundary condition occur in the theory of uni- 
form fields of force. 

A field of force is said to be uniform when the vector E which specifies 
the field strength is the same in magnitude and direction for each point 
of a certain domain D. Taking the direction to be that of the axis of x the 
field strength E may be derived from a potential V of type V Ex by 
means of the equation , 

E = - , > 
ax 

V being an arbitrary constant. This potential V satisfies the differential 
equation /2 r/ 



throughout the domain D. 

Boundary conditions of various types are suggested by physical con- 
siderations. At the surface of a conductor V may have an assigned value. 

At a charged surface -j- may have an assigned value, while there may 

be a surface at which [F] has an assigned value (contact difference of 
potential). 

With boundary conditions of the types that have already been con- 
sidered many interesting problems may be formulated. We shall consider 
only two. 

1-13. Problem 1. To find a solution of d 2 y/dx 2 = which satisfies the 
conditions 

y = when x = and when x = 1 ; [dy/dx] 1 when x ; 

[y] = o- 

The first condition is satisfied by writing 

y = Ax x< g 

= JS(l-a;) x>. 
The condition [y] = gives 

A = B (1 - fl, 



4 The Classical Equations 

and the condition [dy/dx] = 1 gives 

A + B=\. 

.: A=l-f, B = f 
Hence y = 9 (x, ) = a; (1 - ) (a; < 



This function is called the Green's function for the differential expression 
d 2 y/dx 2 , on account of its analogy to a function used by George Green in the 
theory of electrostatics. 

It may be remarked in the first place that a solution of type P + Qx 
which satisfies the conditions y ~ a when x = 0, y = b when x = 1, is given 
by the formula 



Secondly, it will be noticed that </ (a;, ) is a symmetrical function of x 
and ; in other words g (x, ) = g (, #) 

A third property is obtained by considering a solution of d 2 y/dx 2 
which is a linear combination of a number of such Green's functions, for 
example, n 

y= f s g (x, ,), 

s-l 

where f lt / 2 , / 3 , ... are arbitrary constants. The derivative dy/dx drops by 
an amount/j at g l9 by an amount / 2 at 2 , and so on. 
* Let us now see what happens when we increase the number of points 
f i , f 2 > & > an d proceed to a limit so that the sum is replaced by an integral 

y=f l o g(x,$)f()d ...... (A) 



JO 

We find on differentiating that 



= - xf(x) - (1 - x)f(x) = -/(), 

the function / (x) being supposed to be continuous in the interval (0, 1). 
It thus appears that the integral is no longer a solution of the differential 
equation d^/dx 2 = 0, but is a solution of the non-homogeneous equation 



Conversely, if the function / (x) is continuous in the interval (0, 1) a 
solution of this differential equation and the boundary conditions, y = 
when x = and when x = 1, is given by the formula (A); this formula, 



The Green's Function 5 

moreover, represents a function which is continuous in the interval and 
has continuous first and second derivatives in the interval. Such a function 
will be said to be continuous (D, 2), or of class C" (Bolza's notation). 

1-14. Problem 2. To find a solution of d 2 y/dx 2 = and the supple- 
mentary conditions 

[y]= U*-,M 

y = when x = and when x = 1 ; n [dyjdx] + k*y = 0) ' ' 

where s = 1, 2, 3, ... n 1. 

Let y = ^4 8 a; -f # 8 > 5 I < nx < s, 

then the supplementary conditions give B = 0, ^4 n -f J5 n 0, 



n 



?! = 0, 7i (A 2 - A,) -f 2 (AJn -f ^) = 0, 



(A 3 - A,) | + 3 - J? 2 = 0, n (A 3 - AJ + k* (2A 2 /n + B 2 ) = 0, 

H 



nB 3 = (^! + ^4 2 -f ... A 3 ) - sA 8 , 
n 2 (A M - A 8 ) + P (A l + A 2 + ... ^ f ) = 0, 

** ( A *+i - 2 ^ + A *-i) + &A B =0, s > 1. 

This difference equation may be solved by writing k 2 = 2n 2 (1 cos 6), 

.-. ^ 8 = A! cos ( - 1) + K sin (s - 1) 0, 
where K is a constant to be determined. Now 

A 2 = Ai + 2A l (cos 6 - 1) = A l (2 cos - 1), 
therefore K sin = A l (cos 1), 



_ . A sin sQ sin 5 

and so A s = A l - 



The condition = A n -f B n is satisfied if nO = r-n, where r is an integer. 
In the limit when n = oo this condition becomes 

k = lim 2n sin = TTT, 



6 The Classical Equations 

and this is exactly the condition which must b'e satisfied in order that the 
differential equation 



may possess a non- trivial solution which satisfies the boundary conditions 
y = when x = and when x = 1. The general solution of this equation is, 

in fact, ~ , r\ i 

y " cos kx -f Q sin to, 

where P and $ are arbitrary constants. To make y = when # = we 
choose P = 0. The condition y = when x = 1 is then satisfied with Q = 
only if sin & ^ 0, i.e. if k = r?r. 

The exceptional values of P of type (rTr) 2 are called by the Germans 
" Eigenwerte " of the differential equation (B) and the prescribed boundary 
conditions. A non-trivial solution Q sin (kx) which satisfies the boundary 
conditions is called an "Eigenfunktion." These words are now being used 
in the English language and will be needed frequently in this book. To 
save printing we shall make use of the abbreviation eit for Eigenwert and 
eif for Eigenfunktion. The conventional English equivalent for Eigenwert 
is characteristic or. proper value and for Eigenfunktion proper function. 

The theorem which has just been discussed tells us that the differential 
equation (B) and the prescribed boundary conditions have an infinite 
number of real eits which are all simple inasmuch as there is only one type 
of eif for each eit. The eits are, moreover, all positive. 

The quantities & r 2 = ( 2n sin -- j 

may be regarded as eits of the differential equation d 2 y/dx* = and the 
preceding set of boundary conditions. These eits are also positive, and in 
the limit n ->oo they tend towards eits of the differential equation (B) and 
the associated boundary conditions. The solution y corresponding to k is, 
for s 1 < nx < 5, 



TT\ [f S\ ( . STTT . S - 1 . nrX 1 . sr-rrl 
)\(x -sin ---- sin -- -f - sin , ...... (C) 

nl [\ n> \ n n J n n \ v ' 

and it is interesting to study the behaviour of this function as n -> oo to see 
if the function tends to the limit 

*i-i . 

f ^si 

TTT 

T , ., A A f T7r \ 

Let us write A = A 1 (-~} cosec 

1 \nJ \n 

A A 

^o (x) = r7r sin (rirx), F l (x) = x sin (mx), 

and let us use F (x) to denote the function (C) which represents a potygon 
with straight sides inscribed in the curve y F Q (x). 



A Passage to the Limit 7 

The closeness of the approximation of F (x) to F 1 (x) can be inferred 
from the uniform continuity of F Q (x). 

Given any small quantity 6 we can find a number n (e) such that for 
any number n greater than n (c) we have the inequality 



r. (*)-*('- 



< 



< 



for any point x in the interval s 1 < nx < s and for any value of s in the 
set 1, 2, 3, ... n. In particular 

- 1\ _ /s\ 
n ) \n) 

Now F (x) = F Q (-} + (s- nx) \F Q ( S ~ l ] - F (-][ (s- Knx< s). 
\n/ [_ \ n / \n/ j 

Therefore | F (x) - F Q (x) \ < e + c. 

On the other hand 

IV ( / y\ V I f y\ I I V { r r\ 
^0 \ x ) ~ * I \ X ) I ~ I ^0 \ X ) 

Therefore < A 1 \-- cosec 1 

IV 



| ^ (a;) - ^ (a;) | < 2e + ^ cosec ~~ - 

But when is given we can also choose a number m (e) such that for 
n > m (e) we have the inequality 



A [TTT TTT _ "1 I 

, cosec ----- 1 

l [_n n J I 



< . 



Consequently, by choosing n greater than the greater of the two 
quantities n (e) and m (e), if they are not equal, we shall have 

\F(x)~ F, (x) | < 36. 

This inequality shows that as n -> oo, F (x) tends uniformly to the limit 
F, (x). 

This method of obtaining a solution of the equation 

2 +*- 

from a solution of the simpler equation d 2 y/dx 2 = by a limiting process, 
can be extended so as to' give solutions of other differential equations and 
specified boundary conditions, but the question of convergence must always 
be carefully considered. 

1*15. Fourier's theorem. It seems very nat.ural to try to find a solution 
of the equation 



and a prescribed set of supplementary conditions by expanding / (x) in a 



8 The Classical Equations 

d?ii 

series of solutions of - + k*y = and the prescribed supplementary con- 
ditions, because if f(x)=Xb n sinnx (A) 

the differential equation is formally satisfied by the series 

~ , sin nx 
y.S&.i-, 

and if the original series is uniformly convergent the two differentiations 
term by term of the last series can be justified. When the f unction /(#) 
is continuous it is not necessary to postulate uniform convergence because 
Lusin has proved that if the series (A) converge at all points of an interval 
/ to the values of a continuous function/ (x) then the series (A) is integrable 
term by term in the interval 7. Unfortunately it has not been proved that 
an arbitrary continuous function can be expanded in a trigonometrical 
series. Indeed, we are faced with the question of the possibility of expanding 
a given function / (x) in a trigonometrical series of type (A). This question 
is usually made more definite by stipulating the range of values of x for 
which the representation of / (x) is required and the type of function/ (x) 
to which the discussion will be limited. A mathematician who starts out 
to find an expansion theorem for a perfectly arbitrary function will find 
after mature consideration that the programme is too ambitious*, as there 
are functions with very peculiar properties which make trouble for the 
mathematician who seeks complete generality. It is astonishing, however, 
that a function represented by a trigonometrical series is not of an exceed- 
ingly restricted type but has a wide degree of generality, and after the 
discussions of the subject by the great mathematicians of the eighteenth 
century it came as a great surprise when Fourier pointed out that a 
trigonometrical series could represent a function with a discontinuous 
derivative, and even a discontinuous function if a certain convention were 
adopted with regard to the value at a point of discontinuity. In Fourier's 
work the coefficients were derived by a certain rule now called Fourier's 
rule, though indications of it are to be found in the writings of Clairaut, 
Euler and d'Alembert. In the case of the sine-series the rule is that 



2 [ w 
b n = - / (x) sin nxdx, 

it Jo 



and the range in which the representation is required is that of the interval 
(0 < x < TT). When the range is (0 < x < 2ir) and the complete trigono- 
metrical series * 

/ (x) = $0o + 2 a n cos nx 

n-l 

00 

+ S b n sinnx 

n-l 

* For the history of the subject see Hobson's Theory of Functions of a Real Variable and Burk- 
hardt's Report, Jahresbericht der Deuteehen Math. Verein, vol. x (1908). 



Fourier Constants 9 

is to be used for the representation, Fourier's rule takes the form 

a n = - / (x) cos nxdx, 

TT JQ 
1 f 2tr 

b n = - / (x) sin nx dx, (B) 

TT Jo 

and the coefficients a n , b n are called the Fourier constants of the function 



Unless otherwise stated the symbol/ (x) will be used to denote a function 
which is single-valued and bounded in the interval (0, 2n) and defined out- 
side this interval by the equation / (x -f 277) = / (x). 

For some purposes it is more convenient to use the range ( TT < u < TT) 
and the variable u = 2?r x. If / (x) = F (u) the coefficients in the ex- 
pansion of F (u) in a trigonometrical series of Fourier's type are given by 
formulae exactly analogous to (B) except that the limits are TT and TT 
instead of and 2?r. 

The advantage of using the interval ( TT, TT) instead of the interval 
(0, 277-) is that if F (u) is an odd function of u, i.e. if F ( u) = F (u), the 
coefficients a n are all zero, and if F (u) is an even function of u, i.e. if 
F ( u) F (u), the coefficients b n are all zero. In one case the series 
becomes a sine-series and in the other case a cosine-series. 

The possibility of the expansion of / (x) in a Fourier series is usually 
established for a function of limited variation*, that is a function such that 
the sum n _ 1 

s I /(*.)-/(*.) I 

s~0 

is bounded and < N 9 say, for all sets of points of subdivision x l9 x 2y ... a? n-1 
dividing the interval (0, 2n) up into n parts and for all finite integral values 
of n. Such a function is also called a function of limited total fluctuation 
and a function of bounded variation. 

In addition to this restriction on / (x) it is also supposed that the 
integrals in the expressions for the coefficients exist in the ordinary sensef. 
In the case when the integral representing a n is an improper integral it is 
assumed that the integral 

2 (C) 



is convergent. If x is any interior point of the interval (0, 27r) it can be 
shown that when the foregoing conditions are satisfied the series is con- 
vergent and its sum is c 

Mm 



* Whittaker and Watson's Modem Analysis, 3rd ed. p. 175. 

f That is, in the Riemann sense. There are corresponding theorems for the cases in which other 
definitions of integral (such as those of Stieltjes and Lebesgue) are used. 



10 The Classical Equations 

when the limits oif(x e) exist, i.e. with a convenient notation 
H/(* + 0) +/(s - 0)] =/(x), say. 

When the function / (x) is continuous in an interval (a < x < /?) con- 
tained in the interval (0, 2n), is of limited variation in the last interval and 
the other conditions relating to the coefficients are satisfied, it can be 
shown * that the series is uniformly convergent for all values of x for which 
-f S < # -' /J 8, where S is any positive number independent of x. 

When the conditions of continuity and limited variation are dropped 
and the function / (x) is subject only to the conditions relating to the 
existence of the integrals in the formulae for the coefficients and the 
convergence of the integral (C), there is a theorem due to Fejcr, which 
states thatf 

/>) = lim i {A + S, (x) + S 2 (x) + ... S, K _, (x)}, 

m*- oo ft' 

in 

where A = |a , A n (x) = a n cos nx + b n sin nx, S m (x)= 2 A n (x). 

-<) 

This means that the series is summable in the Cesaro sense by the simple 
method of averaging which is usually denoted by the symbol (C, 1). 

This is a theorem of great generality which can be used in applied 
mathematics in place of Fourier's theorem. It is assumed, of course, that 
the limits 

lim /(a?+)=/(a? + 0), Km /(*?-)=/(*- 0) 

t-> -> 

exist J. 

1-16. Cesaro's metJwd of summation . Let 



n8 n = s l + 8 2 + ... + s n , 
then, if S n -> S as n -> oo, the infinite series 



(1) 



is said to be summable (C, 1) with a Cesaro sum S. 

For consistency of the definition of a sum it must be shown that when 
the series (1) converges to a sum s, we have 8 = S. To do this we choose 
a positive integer n, such that 

I Sn+p-*n | <. ...... (2) 

for all positive integral values of p. This is certainly possible when s exists 
and we have in the limit 

|*-*n| <*. ...... (3) 

* Whittaker and Watson, Modern Analysis, 3rd ed. p. 179. 

t Ibid. p. 169. 

J When/ (a: -f 0) = / (x - 0) this implies that f(x) is continuous at the point x. 

Bull, des Sciences Math. (2), t, xrsr, p. 114 (1890). See also Bromwich's Infinite Series. 



Fejer's Theorem 11 

Now let v be an integer greater than n and let C m be defined by the 

equation ~ 

vC m+1 = v-m, ...... (4) 

then S v = c^Ui 4- c 2 ^ 2 4- ... 4- c v u v . ...... (5) 

But Cj > c 2 > ... > c v > 0, hence it follows from (2) that 

I c n+l^n+l 4- C n-f2^n+2 ~H 4" C V U V \ < C n4 . 1 , 

i-e. I S, - (q^ 4- C 2 u 2 -f ... 4- c n ^ n ) | < ec n+1 . 

Making v -> oo we see that if S be any limit of $ 

|S-a| <. ...... (6) 

Combining (3) and (6) we find that 

| S - s | < 2e. 

Since e is an arbitrary small positive quantity it follows that S = s and 
so the sequence $ has only one limit s. 

1-17. Fejer's theorem. Let us now write u^ = ^4 , ?/ n+1 = ^4 n (x), then, 
by using the expressions for the cosines as sums of exponentials, it is readily 
found that* , 



where 26 = | a; ^ | . Now the integrand is a periodic function of / of 
jjb^ioil 277, Consequently we may also write 



b^arthermore, since 

l r '^ = /w 4. 2 (w - 1) cos 26 + 2 (m - 2) cos 46 + . . . + 2 cos 2 (m - 1) 9 
sm' 2 

i i-i xu A f |7r sin 2 w0 | 

it fs readily seen that ---.- - 9 = ^TT. 

J Jo msm 2 ^ 

I Writing ^ (0) = / (x 4- 20) + / (* - 20) - 2/ (a;), 

and inaking use of the last equation, we find that 



w ie r <k (0) -> as -> '0. 

How if e is any small positive quantity we can choose a number 8 



The details of the analysis are given in Whittaker and Watson's Modern Analysis 



12 The Classical Equations 

whenever < 9 < 8, and if is independent of m the number 8 may be 
regarded as independent of m. 
Writing for brevity 

sin 2 m6 = m sin 2 9P (0), TT - 2 
and noting that P (6) is never negative, we have 



o 



(9) fa (9) d9 






P P (9) | + x (9) | d9 + p P (9) \ <t> x (9) \ d9 
Jo Js 






< 

Let us now suppose that | / (t) \ dt exists, then 

J TT 

f I & (9) I d9 
Jo 

also exists, and by choosing a sufficiently large value of m we can make 

aem sin 2 8 > f" I J> (9) I d9. 
Jo 

This makes the second integral on the right of (B) less than e, which 
is also the value of the first integral. Therefore 

\S m (x)-f(x)\ <2ae/7r=e; 
consequently S m (x) -> / (x) as m -> oo. 

When/ (x) is continuous throughout the interval ( TT < x < TT) all! ,i 
foregoing requirements are satisfied and in addition / (x) = f (x) ; col & 
quently, in this case, S n (x) -> f (x), and this is true for each point I o 
the interval. \ f 

This celebrated theorem was discovered by Fejer*. The conditional oJ 
the theorem are certainly satisfied when the range ( TT < x < TT) canpbe 
divided up into a finite number of parts in each of which / (x) is bo <mded 
and continuous. Such a function is said to be continuous bit by bit 
(Stiickweise stetig); the Cesaro sum for the Fourier series is then/ (x) at 
any point of the range, / (x) and / (x) being the same except at the points 
of subdivision. 

1-18. ParsevaTs theorem. Let the function / (x) be continuous bit by 
bit in the interval ( TT, ?r) and let its Fourier constants be a n , 6 n ; it 
then be shown that 



[/ (*)? dx = TT [~K 2 + 2 (a n 2 -f 6 n 2 )l ....... (A) 

L n=l J 



Math. Ann. Bd. LVIII, S. 51 (1904). 



ParsevaVs Theorem 



13 



We shall find it convenient to sum the series (A) by the Ces&ro method* 
This will give the correct value for the sum because the inequality 



; -f 



n-l 



n-1 



indicates that the series is convergent. 

To find the sum (C, 1) we have to find the limit of S m where, by a simple 
extension of 1-17 (A), 



S -- 

M ~ 277 _ .^ 



sn* 



29 being equal to | x t \ . 

Since the region of integration can be divided up into a finite number 
of parts in each of which the integrand is a continuous function of x and t, 
the double integral exists and can be transformed into a repeated integral 
in which x and are the new independent variables. The region for which 
6 lies between 6 Q and -f dd, while x lies between X Q and X Q -h dx consists 
of two equal partsf; sometimes two, sometimes one and sometimes none 
of these parts lie within the region of integration. When this is taken 
into consideration the correct formula for the transformation of the 
integral is found to be 



1 fir 

.= H- 

*TTJQ 



n-20 



20) 



fTr 
JW-T 



In the derivation of this result Fig. 1 will 
be found to be helpful. The lines M l M 2y 
M Z M are those on which has an assigned 
value, while N 1 N 2 , JV 3 A 7 4 are lines on which 
has a different assigned value. It will be 
noticed that a line parallel to the axis of t 
meets M 1 M 2 , M 3 M^ either once or twice, while 
it meets N 1 N 29 N^N^ either once or not at all. 

Applying the theorem of 1-17 to (B) we 
get 



-7T 



...... (B) 



7T-20 




7T 



Fig. 1. 



- r f(x)/(x)dx, 

J IT 

and when/ (x) is defined to be/ (x) this result gives (A). 



lim , 

m->oo 



* This is the plan adopted in Whittaker and Watson's Modern Analysis, p. 181. The present 
proof, however, differs from that given in Modern Analysis, which is for the case in which f (x) is 
bounded and integrable. 

t It will be noted that the Jacobian of the transformation has a modulus equal to two. 



14 The Classical Equations 

The theorem (A) was first proved by Liapounoff*; the present investi- 
gation is a modification of that given by Hurwitzf. 

Now let F (x) be a second function which is continuous bit by bit in 
the interval -n < x < 77 and let A n , B n be its Fourier constants. Applying 
the foregoing theorem to F (x) -f / (x) and F (x) f (x), we obtain 

T [F (x) + f (x)Y dx = 77 [i (A + a ) 2 -f S {(4 n + aj 2 -f 

J-7T L W=l 

I" [F (x) - f (x)]* dx - TT [4 Mo - a )2 + S {(4 n - a J 2 + 

./-* L n-1 / 

Subtracting, we obtain the important formula 

I* / (x) F (x) dx = >n \!>A a + 2 (,4 n a n + B n b n ) 

J-1T L W-l 

which is usually called Parseval's theorem, though ParsevaFs derivation 
of the formula was to some extent unsatisfactory. 

In the modern theory, when Lebesgue integrals are used, the theorem 
is usually established for the case in which the functions f (x), F (x), 
[f (x)] 2 and [F (x)] 2 are integrable in the sense of Lebesgue. There is also 
a converse theorem which states that when the series (A) converges there 
is a function / (x) with a n and 6 n as Fourier constants which is such that 
[/ (#)] 2 i s integrable and equal to the sum of the series. This theorem was 
first proved by Riesz and Fischer. Several proofs of the theorem are given 
in a paper by W. H. Young and Grace Chisholm Young J. The theorem has 
also been extended by W. H. Young , the complete theorem being also 
an?> extension of Parseval's theorem. A general form of Parseval's theorem 
has been used to justify the integration term by term of the product of a 
function and a Fourier series. 

ADDITIONAL RESULTS 

1. If the functions / (x), F (x) are integrable in the sense of Lebesgue, and [/(z)] 2 , 
[F (x)] 2 are also integrable in the same sense, then|| 



T 

W J -T 



x)F(t)dt = K^o+ 2 (a n A n H- b n B n ) cos nx - 2 (a n B n - b n A n ) sin nx. 



2. If / (x) is a periodic function of period 2?r which is integrable in the sense of Lebesgue, 
and if g (x) is a function of bounded variation which is such that the integral 

'00 

\g(x)\ dx 
o 

* Comptes Rendus, t. cxxvi, p. 1024 (1898). 

t Math. Ann. Bd. LVII, S. 429 (1903). 

J Quarterly Journal, vol. XLJV I p. 49 (1913). 

^ Comptes Rendus, t. CLV, pp. 30, 472 (1912); Proc. Roy. Soc. London, A, vol. LXXXVII, p. 331 
(1912); Proc. London Math. Soc. (2), vol. xii, p. 71 (1912). See also F. Hausdorff, Math. Zeits. 
Bd. xvi, S. 163(1923). 

|| W. H. Young, Comptes Rendus, t. CLV, p. 30 (1912); Proc. Roy. Soc. London, A, vol. 
LXXXVII, p. 331 (1912). 



Fourier Series 15 

is convergent, then the value of the integral 

r/(x)g(x)dx 
Jo 

may be calculated by replacing / (x) by its Fourier series and integrating formally term by 
term. In particular, the theorem is true for a positive function g (x) which decreases steadily 
as x increases and is such that the first integral is convergent. 

[W. H. Young, Proc. London Math. Soc. (2), vol. ix, pp. 449, 463 (1910); vol. xin, p. 109 
(1913); Proc. Roy^oc. A, vol. xxxv, p. 14 (1911). G. H. Hardy, Mess, of Math. vol. LI, 
p. 186(1922).] 

1-19. The expansion of the integral of a bounded function which is 
continuous bit by bit. If in Parse val's theorem we put 

F (x) = 1 , - TT < x < z, F (x) = 0, z < x < TT, 
we have A^ = - F (x) dx = - dx = - n , 

TT J -n 7T J-TT TT 

A n = - cos nx.dx = [sin nz], 
TT ) - n n-n 

B n = - sin nx.dx [cos nn cos nz], 

TT j _ ff nrr 

and we have the result that 

JZ co ] 

/ (x) dx = |a (z -f 77) -|- 2 - [a n sin nz -\- b n (cos WTT cos nz)]. 
-TT n=-i n 

...... (A) 

Now the function ^a z can be expanded in the Fourier series 

- 1 

o 2 ~ cos mr sin /i^, 
= i 

hence the integral, on the left of (A) can be expanded in a convergent 
trigonometrical series. To show that this is the Fourier series of the 
function we must calculate the Fourier constants. 

n 

dx 



. . a n a () 

H / (2) c ^ s nz = ----- cos mr, 
J v ' n n 



1 [ n [ z 1 ( n 

Now - sin nz dz f (x) dx = ---- cos mr f (x) 

7Tj-n Jrr M J -n 

If 71 " dz p . . 

/ (2) 

TT ]- n J v ' 

1 f* [ s 1 [" 

cos nzdz f (x) dx = dz sin nz f (z) 

TTJ-rr J-n HIT J - n 

-^- 

- f dz f / (x) dx=\* f (x) dx - ! f zf(z) dz 

TT)-* J-n J -n ^ J -n 



OJ J 

TTO,, + 2 - & cos n?r, 



16 The Classical Equations 

by Parseval's theorem. Hence the coefficients are precisely the Fourier 
constants and so the integral of a function which is continuous bit by bit 
can be expanded in a Fourier series. This means that a continuous periodic 
function with a derivative continuous bit by bit can be expanded in a 
Fourier series. 

Proofs of this theorem differing from that in the text are given by 
Hilbert-Courant, Methoden der Mathematischen Physik, Bd. I (1924), and 
by M. G. Carman, Bull. Amer. Math. Soc. vol. xxx, p. 410 (1924). 

It should be noticed that equation (A) shows that when / (x) can be 
expanded in a Fourier series this series can be integrated term by term. 
A more general theorem of this type is proved by E. W. Hobson, Journ. 
London Math. Soc. vol. n, p. 164 (1927). 

Fourier's theorem may be extended to functions which become infinite 
in certain ways in the interval (0, 277). When the number of singularities is 
limited the singularities may be removed one by one by subtracting from 
/ (x) a simple function h s (x) with a singularity of the same type. This 
process is continued until we arrive at a function 

(*)=/(*)- 2 h.(x) 

S-l 

which does not become infinite in the interval (0, 277). The problem then 
reduces to the discussion of the Fourier series associated with each of the 
functions h s (x). 

1-21. The bending of a beam. We shall now consider some boundary 
problems for the differential equation d*yfdx* = 0, which is the natural one 
to consider after d*y/dx 2 = from the historical standpoint and on account 
of the variety of boundary conditions suggested by mechanical problems. 

The quantity y will be regarded here as the deflection from the equi- 
librium position of the central axis of a long beam at a point Q whose 
distance from one end is x. The beam will be assumed to have the same 
cross-section at all points of its length and to be of uniform material, also 
the deflection at each point will be regarded as small. The physical pro- 
perties of the beam needed for the simple theory of flexure are then 
represented simply by the value of a certain quantity B which is called 
the flexural rigidity and which may be calculated when the form of the 
cross-section and the elasticity of the material of the beam are known. 
We are not interested at this stage in the calculation of B and shall conse- 
quently assume that the value of B for a given beam is known. The funda- 
mental hypothesis on which, the theory is based is that when the beam is 
bent by external forces there is at each point x of the central axis a resisting 
couple proportional to the curvature of the beam which just balances the 
bending moment introduced by the external forces. When the flexure 
takes place in the plane of xy this resisting couple has a moment which 



S+dS 




Bending of a Beam 17 

can be set equal to JBd*y/dx 2 and the fundamental equation for the bending 
moment is M = Bdty(da . tm 

The origin of the bending moment will 
be better understood when it is remarked 
that the bending moment M is associated 
with a transverse shearing force 8 by the 
equation 

- s = dM/dx. 

When the beam is so light that its Fig. 2. 

weight may be disregarded, this shearing force S is constant along any 
portion of the beam that does not contain a point of support or point of 
attachment of a weight. If we have a simple cantilever OA built into a 
wall at O and carrying a weight W at the point B the shearing force 8 is 
zero from A to B and is W from B to 0, 
while M is zero from A to B and equal 
to Wx between B and 0. At the point 
the fact that the beam is built in or 
clamped implies that ?/== and dy/dx = 0, 
consequently the equation 

Bd*y/dx* = - W x + W b 
gives y = - W x*/6B + Wbx 2 /2B. 



O 



B 



W 



-a- 



This holds for x < b. For x > b the differential equation for y is 

Bd*y/dx* = 0, 
and so y mx + c. 

The quantities y and dy/dx are supposed to be continuous at B and so 
we have the equations 

mb + c = W b*/3B, m = PF6 2 / 25 

which give c == Wb 3 /6B. The deflection of B is Wb 3 /3B and is seen to be 
proportional to the force W, The deflection of A is also proportional to W. 

1-22. Let us next consider the deflection of a beam of length I which 
is clamped at both ends x = 0, x = I and which carries a concentrated 
load W at the point x = . 

We have the equations 



S = I 7 + ff, S = T, say, 

- M = (T+ W" )x + ^= -Bd 2 y/dx*, - M = Tx 
(T 7 + IF) x 2 + NX - - JWy/ete, ^T 7 ^ 2 



$(T+W)x*+ $Nx* = - By, 



W + N = - Bd 2 y/dx 2 , 

N)(x-l) = - Bdy/dx, 
(a: - /) 



18 The Classical Equations 

where T and N are constants to be determined. M has been made con- 
tinuous at x = , but we have still to make y and dy/dx continuous. This 
gives the equations 

\(W&- T/ 2 )- Wlg + Nl, 
i (Wf + TZ) = j (_ Pf + T) Pf + prj (2f _ i) + 

Therefore Tl* = W* (2f - 3Z), ZW = Iff (2Z - Z 2 - 

(Z - ) 2 [a; (Z + 2f) 



This solution will be written in the form By = TFgr (#, f ) and the 
function gr (x, ^) will be calle.d a Green's function for the differential ex- 
pression d*yldx* and the prescribed boundary conditions. 



If By= 

it is found on differentiation that y is a solution of the differential equation 



the function w (x) being supposed to be continuous in the range (0, /). 
Thi^ solution corresponds to the case of a distributed load of amount wdx 
for a length dx. -When w is independent of x the expression found for y is 

in 

By=^x*(l-x)*. 

It should be noticed that the Green's function g (x, f ) is a symmetrical 
function of x and y, its first two derivatives are continuous at x = f , but 
the third derivative is discontinuous, in fact 



The reactions at the ends of the clamped beam with concentrated load 
are found by calculating the shear S. When x < we have 

S = W (I - ^) 2 (Z + 2f )/Z, 

and this is equal in magnitude to the reaction at x == 0. The reaction at 
x = I is similarly R = W (31 - 2f) f 2 /Z'. 

The deflection of the point x = f is 

% = W? (I - WIWB, 

when - Z/2 this amounts.to W7 S /192J3. 

In the case of the uniformly loaded beam the reactions at the ends are 
respectively W and | W as we should expect. The deflection of the middle 
point is W Z 4 /384. 



The Green's Functions 19 

1*23. When the beam is pin -jointed at both ends, M is zero there and 
the boundary conditions are 

y = 0, d 2 y/dx 2 = for x = and x = a. 

When there is a concentrated load IF at x = and the beam is of 
negligible weight, the solution is By Wk (x, ), where 
& (x, f) = x (a - ) (x 2 + 2 - 
= g(a- x) (x 2 -f 2 - 
The reactions at the supports are 

J? = W (1 - |/a) at a = 0, 
J^a = W^/fl at a: = a. 

As before, the deflection corresponding to a distributed load of density 
w (x) is 



and when w is constant 

By = w (x* - 2ax* -f a 3 #)/24. 
The reactions at the supports are in this case 

7? = wcl n 

./TO " o a u x v/j 

2i 

T> W(l 

jKa at* x === a* 

In the case of a beam of length Z clamped at the end x = and pin- 
jointed at the end x = Z, the solution for the case of a concentrated load 
W at x = | is 

The deflection at # = f is now 

y = Tf3 (j _ ^2 (4jj _ |)/12 

when | = Z/2, and the reaction at # = Z is 

If, on the other hand, we consider a beam which is clamped at the end 
x = but is free at the end x = Z except for a concentrated load P which 
acts there, we have, at the point x , 

while at the point x I 



By l = 
Hence R = Wy f: jy l . 

If the original beam is acted on by a number of loads of type W we 
have, for the reaction at the end x = Z, 



20 The Classical Equations 

On account of this relation the curve (I) is called-the " influence line" 
of the original beam. Much use is made now of influence lines in the theory 
of structures*. 

There are three reciprocal theorems analogous to g (x, ) = g (, x) 
which are fundamental in the theory of influence lines. These theorems, 
which are due to Maxwell and Lord Rayleigh, may be stated as follows: 

Consider any elastic structure with ends fixed or hinged, or with one 
end fixed and the other hinged, to an immovable support, then 

(1) The displacement at any point A due to a load P applied at any 
point B is equal to the displacement at B due to the same load P placed 
at A instead of B. 

(2) If the displacement at any point A is prevented by, a load P at 
A with displacement y B at B under a load Q, and alternatively if a load 
Q l at B prevents displacement at B with displacement y A at A under a 
load P, then if y A = y R , P must equal Q l . 

(3) If a force Q acts at any point B producing displacements y B at 
B and y A at any other point A, and if a second force P is caused to act at 
A but in the opposite direction to Q reducing the displacement at B to 
zero, then Q/P = y A /yB* 

In these three relationships it is supposed that the displacements are 
in the directions of the acting forces. 

Proofs of these relations and some applications will be found in a paper 
by C. E. Larard, Engineering, p. 287 (1923). 

1-24. Let us next consider a continuous beam with supports at A, B 
and C. The bending moment M at any point in AB or BC is the sum of 
bending moment M l of a beam which is pin-jointed at ABC and of the 
moment M 2 caused by the fixing moments at the supports. Let us take B 
as origin and let 1 2 denote the length BC. 

For a beam which is pin-jointed at B and C we have 

M 1 = \w (I 2 x - x 2 ), 

while for a weightless beam with fixing moments M B , M c at B and C 
respectively, we have 

M 2 = - M B - (M c - M B ) x/l 2 . 

Hence M = B 2 d 2 y/dx 2 = \w 2 l 2 x - \w 2 x 2 - M B - (M c - M B ) x/l 2 . 
Integrating, we have 

B 2 dy/dx = w 2 l 2 x 2 - w 2 x*/$ - M B x - x 2 (M c - M B )/21 2 - B 2 i B , 
where IB is the value of dy/dx at x = 0. Integrating again, 

B 2 y = W 2 l 2 x 3 /l2 - w 2 x*/24: - x 2 M B /2 - x 9 (M c - M B )/6l 2 - B 2 i B x. 
When x - 1 2 , y = - y 2 , say 

6B 2 i B = w 2 l 2 * - 2M B 1 2 - M C 1 2 + M^yz. 

* See especially Spofford, Theory of Structure*; D. B. Steinman, Engineering Record (1916); 
G. E. Beggs, International Engineering, May (1922). 



The Equation of Three Moments 21 

Similarly, by considering the span BA, taking B again as origin, but 
in this case taking x as positive when measured to the left, 

Eliminating i& we obtain the equation 

B^ (M A + 2M B ) + BJt (M c + 2M B ) - J (^A 3 S 2 + wJfBJ 

+ 6 (B 2 l l ~ l y l + B^l^y^. 

This is the celebrated equation of three moments which was given in 
a simpler form by Clapeyron* and subsequently extended for the general 
case by Heppelt, WeyrauchJ, Webb and others ||. 

The reaction at B is the sum of the shears on the two sides of B and is 

therefore 7 ,, ,, 7 ,, ,, 

% = W 2*2 + MB- MC ^ wJi M R - M A 

Similarly for the other supports. 

1-25. When a light beam or thin rod originally in a vertical position 
is acted upon by compressive forces P at its ends (Fig. 4) | 
the equation for the bending moment is 

M = Bd 2 y/dx 2 = - Py, 
or d 2 y/dx 2 + k z y = 0, 

where k 2 = P/B; 

and if y = when x and when x = a, the solution is 
y = A sin kx, where sin ka = or A = 0. 

If ak < 77, the analysis indicates that A = 0. A solution 
with A becomes possible when ak = TT. The corresponding 
load P = B7T 2 /a 2 is called Euler's critical load for a rod pinned 
at its ends. When P is given there is a corresponding critical 
lejijgth a = P^/B^Tr. 

To obtain these critical values experimentally great care 
must be taken to eliminate initial curvature of the rod and 
bad centering of the loads. The formula of Euler has been confirmed by 
the experiments of Robertson. In general practice, however, the crippling 
load PC is found to be less than the critical load P given by Euler's formula, 
and many formulae for struts have been proposed. For these reference 
must be made to books on Elasticity and the Strength of Materials. 

In the case of a strut clamped at both ends there is an unknown couple 
M Q acting at each end. The equation is now 

Bd*y/dx 2 + Py=M , 

* E. Clapeyron, Comptes Retidus, t. XLV, p. 1076 (1857). 

t J. M. Heppel, Proc. lust. Civil Engineers, vol. xix, p. 625 (1859-60). 

J Weyrauch, Theorie der continuierlichen Trdger, pp. 8-9. 

R. R. Webb, Proc. Camb. Phil Soc. vol. vi (1886). (Case B l 4= B 2 .) 

|| M. Levy, Statique graphique, t. n (Paris, 1886). (Case y l 4= 0, y z 4 s 0.) 




22 The Classical Equations 

and the solution is of type 

Py = MQ -f a cos kx -\- j$ sin &#. 

The boundary conditions y = 0, dy/rfa = at a: = a and # = give 
0=0, a = M Q , MQ sin &a = 0, M (1 cos ka) = 0. 

Hence either Jfef = or sin (to/2) = 0. The critical load is now given by 
the equation ka = 2?r and is P = 4Brr 2 /a 2 . 

When the load P reaches the critical value the rod begins to buckle, and 
for a discussion of the equilibrium for a load greater than P Q a theory of 
curved rods is needed. 

In the case of a heavy horizontal beam of weight w per unit length and 
under the influence of longitudinal forces p at its ends, the equation 
satisfied by the bending moment M is 



where k* = P/B. 

If M = M when x = and M = Mj when a; = a, the solution of the 
differential equation is 



v^ M) sin ka = ( 2 "" ^o ) s i n ^ ( a ~" x ) + ( 7T 2 ~ -^i ) s * n ^ 
Let us assume that M Q and Jf 1 are both positive and write 

M = , sin 2 0, Jkf i = , sin 2 ^, kx = a. k (a x) B, ka = a + B. 
fc 2 k* r 

The equation which determines the points of zero bending moment 
(points of inflexion) is 

sin (a -f 0) = cos 20 . sin j8 -f cos 2(f> . sin a. 

We shall show that if a and are both positive this equation implies that 
a-f/3> 2 (# + </>) and so determines a certain minimum length which must 
not be exceeded if there are to be two real points of inflexion. 

Let us regard 6 and <f> as variable quantities connected by the last 
equation and ask when + </> is a maximum. Writing z = B -f <f> we have 

-T| = 1 + -~f , = sin 28 sin B -f- sin 2<A sin a ^ , 

au au au 



*z d*<f> 2sinj8 f 

b" 2 = "^ ^ ~ ~ - ^T7T7 c 
O* dO* sm a sm 2 2(f>[_ 



ft rt/ . 
cos 2 sm 2 - sin 



When 2$ = a, we have 2<f> = j8, and these values of 6 and < give a zero 
value of JQ and a negative value of ^ , they therefore give us a maximum 

value of 2, and so for ordinary values of 6 and <f> we have the inequality 

2 (0 + <) < a -f j8. 



Stability of a Strut 23 

The position of the points of inflexion is of some practical interest 
because, in the first place, A. R. Low* has pointed out that instability is 
determined by the usual Eulerian formula for a pin- jointed strut of a length 
equal to the distance between the points of inflexion, if these lie on the 
beam, and secondly, if any splicing is to be done, the flanges should 'be 
spliced at one of the points where the bending moment is zerof. 

When P is negative and so represents a pull we may put p 2 = - P/B, 
and the solution is 

( ~ 2 ~^~ M } sinh pa = ( -f M Q } sinh p (I x) -f ( 4- M* ) sinh vx. 
\p* / \p ) N \p 2 L J r 

If we write 

2w ,- 2w 

M = sinh- 0, MI "- sinh 2 ^6, px = a, p (a - x) = )8, pa = a + fi, 
p p 

a value of x for which M == is determined by the equation 
sinh (a -f- jS) = cosh 20 sinh ^3 -f cosh 2^> sinh a. 
This equation implies that 

a -f j3 > 2 (0 -f (/>). 

For a continuous beam acted on by longitudinal forces at the points of 
support there is an equation analogous to the equation of three moments 
which is obtained by a method similar to that used in obtaining the ordinary 
equation of three moments. We give only an outline of the analysis. 

Case 1. k*=P 2 /B 2 , EG = 6, 2/3 = bk, y A = VB = y c =* 0, 




where 



* .4ero?iaMftcaZ Journal, vol. xvin, p. 144 (April, 1914). See also J. Perry, Phil Mag. (March, 
1892); A. Morley, ibid. (June, 1908); L. N. G. Filon, Aeronautics, p. 282 (Sept. 1919). 
t H. Booth, Aeronautical Journal, vol. xxiv, p. 563 (1920). 



24 The Classical Equations 

There is a corresponding equation for the bay BA which is of length a, 
if h 2 = Pi/Bi , 2a = ah, the equation of three moments has the form 

aM A bM c f,n\ , OTIT ! a JL t \ _. * JL /m' ^i a3 / / \ _L 

-^-/(a)-f -g- /(/?) -f 2 ^ j^ '0 ()+ ^"^(P)f = 4^ ^( a ) + 

Case 2. P < 0. The corresponding equations are 

Pj/JJj, 2a = ah, 2/? = 6i, 

O (a) + I O (j8)J 



r, . . 3 1 - 2a cosech 2a A , . 3 2a coth 2a 1 
*<) = 2-~ --, - -, OW-i - ^i 

V (a) = 

The functions/ (a), P (a), etc., have been tabulated by Berry* who has 
also given a complete exposition of the analysis. These equations are much 
used in the design of airplanes built of wood. 

EXAMPLES 

1. Find the crippling load for a rod which is clampod at one end and pinned at the other. 

2. Prove that jn the case of a uniform light beam of length a with a concentrated load 
W at x = | the solution can be written in the form 

2Wa I . nx . TT 1 . 2nx . 2n{ \ 
M = - 1 sin sin --- h ^ sin sin +...), 
7T 2 \ a a 2 2 a a ) 

TTX . TT 1 

a~ 8m a + 2* 

when the beam is pinned at both ends. The corresponding formulae for a uniformly dis- 
tributed load are 

, . 4tW f . TTX 1 . 3irX 1 . 5nX \ 

w(x) = - (sin -f O sm ---- 1- _ sin +...), 
TT \ a 3 a 5 a ) 

,. 

M = 



va 2 ( . nx 1 . 3nx 1 . 57ra; \ 
-- sm ---- h jr- sin --- h ^> sin --- h ... ) , 

T 3 \ a 3 3 a 5 3 a / 

4wa* f . TTX 1 . STTX 1 . STTX \ 

y = D . sin --- h 5- sm -f- K5 sin --- -f ... . 

Bn 5 \ a 3 a 5 5 a / 



[Timoshenko and Lessells Applied Elasticity, p. 230.] 

3. Find the form of a strut pinned at its ends and eccentrically loaded at its ends with 
compressional loads P. 

4. The Green's function for the differential expressionf 



* Trans. Roy. Aeronautical Soc. (1919). The tables are given also inPippard and Pritchard*a 
Aeroplane Structures, App. I (1919). 

f Examples 4-6 are taken from a paper by A. Myller, 2$as. Oottingen (1906). 



End' Conditions 25 

and the end conditions u (0) - u (I) = u' (0) - u' (1) - is 



~ fl\*-\ &> *M- 

^ _ [ l *jk 

(z)> P- Jo <*")' 



where 



5. If in the last example the boundary conditions are u (0) = u' (0) = u" ( 1 ) = u"' ( 1 ) = 
the Green's function is 

* ' * ~ 4 



6. When the end conditions are u (0) = w" (0) = u (1) =. u" (1) = 0, the Green's 
function is (#, ( ), where 



- (a - 4)3 + 4y)*f - (0 - 2y)(x + f ) - y. 



1-31. jFVee undamped vibrations. Whenever a particle performs free 
oscillations in a straight line under the influence of a restoring force pro- 
portional to the distance from a fixed point on the line the equation of 

motion is 



mx = - 



where m is the mass of the particle and X\L is the restoring force. Writing 
M = 4m, we have 



an equation which has already been briefly considered. The general 
solution is 



gn 



where ^4 and JS are arbitrary constants. Writing k = 27m, A = a sin 
B = a cos we have 

x = a sin (27m -f 0). 



The quantity a specifies the amplitude, n the frequency and 27fnt -f 
the phase of the oscillation. The angle gives the phase at time t = 0. 
The period of vibration T may be found from the equations 

kT - 277, nT = 1. 

This type of vibration is called simple-harmonic vibration because it is 
of fundamental importance in the theory of sound. The vibrations of solid 
bodies which are almost perfectly rigid are often of this type, thus the 
end of a prong of a tuning fork which has been properly excited moves in 
a manner which may be described approximately by an equation of this 



26 The Classical Equations 

type. The harmonic vibrations of the tuning fork produce corresponding 
vibrations in the surrounding air which are of audible frequency if 

24 < n < 24000. 
The range of frequencies used in music is generally 

40 < n < 4000. 

The differential equation (I) may be replaced by two simultaneous 
equations of the first order 

x + ky=0, y-kx=0 (II) 

which imply that the point Q with rectangular co-ordinates (x, y) moves 
in a circle with uniform speed ka. We have, in fact, the equation 

xx 4- yy = 0, 

which signifies that x 2 -f- y 2 is a constant which may be denoted by a 2 . 
There is also an equation 

#2 -f I/* = k* (3.2 _j_ y 2) = 2 a 2 ? 

which indicates that the velocity has the constant magnitude ka. The 
solution of the simultaneous equations may be expressed in the form 

x = a cos a, y = a sin a, 
where a Zrrnt 4- 6 ~ ; 

Simultaneous equations of type (II) describe the motion of a particle 
which is under the influence of a deflecting force perpendicular to the 
direction of motion and proportional to the velocity of the particle. The 
equations of motion are really 

x -f- ky 0, y kx = 0, 

but an integration with respect to t and a suitable choice of the origin of 
co-ordinates reduces them to the form (II). The equations may also be 

written in the form 7 , 

u + kv = 0, v leu = 0, 

where (u, v) are the component velocities. 

If the deflecting force mentioned above is the deflecting force of the 
earth's rotation the deflection is to the right of a horizontal path in the 
northern hemisphere and to the left in the southern hemisphere. If the 
angle <f> represents the latitude of the place and o> the angular velocity of the 
earth's rotation, the quantity k is given by the formula 

k 2a> sin (f>. 

When the resistance of the air can be neglected, the suspended mass M 
of a pendulum performs simple harmonic oscillations after it has been 
slightly displaced from its position of equilibrium. The vertical motion is 
now so small that it may be neglected and the acceleration may, to a first 




Simple Periodic Motion 27 

approximation, be regarded as horizontal and proportional to the horizontal 
component of the pull P of the string. We thus have the equation of motion 

Mix = - Px = - Mgx, 

where / is the length of the string and g the acceleration of 
gravity. The mass of the string is here neglected. With this Q 
simplifying assumption the j endulum is called a simple pendu- 
lum. In dealing with connected systems of simple pendulums 
it is convenient to use the notation (I, M) for a simple pendu- 
lum whose string is of length I and whose bob is of mass M 
(Fig. 5). 

If the string and suspended mass are replaced by a rigid 
body free to swing about a horizontal axis through the point 
0, the equation of motion is approximately 

19 = - MgliO, 

where 7 is the moment of inertia of the body about the horizontal axis 
through and h is the depth of the centre of mass below the axis in the 
equilibrium position in which the centre of mass is in the vertical plane 
through 0. Writing Mhg = Ik 2 the equation of motion becomes 

+ ]*0 - 0, 

and the period of vibration is 27r/k, a quantity which is independent of 
the angle through which the pendulum oscillates. 

This law was confirmed experimentally by Galileo, who showed that 
the times of vibration of different pendulums were proportional to the 
square roots of their lengths. The isochronism of the pendulum for small 
oscillations was also discovered by him but had been observed previously 
by others. When the pendulum swings through an angle which is not 
exceedingly small it is better to use the more accurate equation 

+ 2 sin 0=0, 

which may be derived by resolving along the tangent to the path of the 
centre of gravity G or by differentiating the energy equation 

= Mgh (cos - cos a), 



which is written down on the supposition that the velocity of is zero 
when = a. With the aid of the substitution 

sin (|0) = sin (Ja) sin</>, 
this equation may be written in the form 

^2^ p [i _ sin 2 \a sin 2 <]. 

As varies from a to a, <f> varies from ^ 5 > an( ^ so 



28 The Classical Equations 

swing from one extreme position (0 = a) to the next extreme position 
(0 = ) is 

2 (1 - sin 2 \a sin 2 </)"* d*j>. 

When a is small the period T is given approximately by the formula 
kT = 27r(l -f Jsin 2 J) 

and depends on a, so that there is not perfect isochronism. 

This fact was recognised by Huygens who discovered that perfect 
isochronism could theoretically be secured by guiding the string (or other 
flexible suspension) with the aid of a pair of cycloidal cheeks so as to make 
the centre of gravity describe a cycloidal instead of a circular arc. This 
device has not, however, proved successful in practice as it introduces 
errors larger than those which it is supposed to remove*. More practicable 
methods of securing isochronism with a pendulum have been described by 
Phillipsf. 

1-32. Simultaneous equations of type 

Lx -j- My -f Lm 2 x = 0, 
MX + Nij -f Nn*y = 0, 

in which L, M, N y m y n are constants, occur in many mechanical and 
electrical problems. When the coefficient M is zero the co-ordinates x and 
y oscillate in value independently with periods Sir/m and 27r/n respectively, 
but when M = the assumption 

x = p 2 MAe lpt , y = LA (m 2 - p 2 ) e ipt 
gives the equation 

p* (1 - y 2 ) - p 2 (m 2 4- n 2 ) + m 2 n 2 = 0, 
where y 2 = M 2 /LN. 

This quantity y is called the coefficient of coupling! . 
When m =5^ ?i we have - . 

V 2 D 4 
7)2_ m 2 == Yf 

* p 2 -n 2 ' 

and when y is small the value of p which is close to m is given approximately 

by the equation 

>p2 _ m 2 = -JT _ p 2 say . 
* m 2 - n 2 r > j 

A simple harmonic oscillation of the #-co-ordinate, with a period close 
to the free period 2-jT/m, is accompanied by a similar oscillation of the 

* See R. A. Sampson^ article on " Clocks and Time-Keeping" inDictionary of Applied Physics, 
vol. m. 

t Comptes Rendus, t. oxn, p. 177 (1891). 

J See, for instance, E. H. Barton and H. Mary Browning, Phil. Mag. (6), vol. xxxiv, p. 246 
(1917). 



Simultaneous Linear Equations 29 

y-co-ordinate with the same period but opposite phase. The amplitude of 
the ^-oscillation is proportional to y 2 . Now let p 1 be the greater of the two 
values of p. If m > n we have p l > m but if m < n we have p 2 < m. The 
effect of the coupling is thus to lower the frequency of the gravest mode of 
vibration and to raise the frequency of the other mode of simple harmonic 
vibration. If m = n the equation for p 2 gives 

p 2 = m 2 yp 2 , 

and the effect of the coupling is to make the periods of the two modes 
unequal. In the general case we can say that the effect of the coupling 
is to increase the difference between the periods. The periods may, in fact, 
be represented geometrically by the following construction : 

Let OA, OB represent the squares of the free periods, the points 0,A,B 
being on a straight line. Now draw a circle F on AB as diameter and let 
a larger concentric circle cut the line OA B in U and V \ the distances OC7, 
O V then represent the squares of the periods when there is coupling. If a 
tangent from to the circle F touches this circle at T and meets the larger 
circle in the points M and L the coefficient of coupling is represented by 
the ratio TL/TO (Fig. 6). 

So long as lies outside the larger circle it is evident that the difference 
between the periods is increased by the coupling, but when y > 1 the point 
lies within the larger circle and the difference between the periods de- 
creases to zero as the radius CU of this circle increases without limit. There 
is thus some particular value of the coupling for which the difference 
between the periods has the original value, both periods being greater than 
before. 

When y = 1 the equations of motion may be written in the forms 

Lx + My + Lm 2 x = 0, Lx + My + Mn 2 y = 0, 

and imply that Lm 2 x = Mn 2 y. There is 
now only one period of vibration. The 
cases y> \ are not of much physical 
interest as the values of the constants 
are generally such that M 2 < LN, this 
being the condition that the kinetic 
energy may be always positive. 

Equations of the present type occur 
in electric circuit theory when resist- 
ances are neglected. In the case of a 
simple circuit of self-induction L and lg * 

capacity C furnished, say, by a Ley den jar in the circuit, the charge Q 
on the inside of the jar fluctuates in accordance with the equation 

= 




30 The Classical Equations 

when the discharge is taking place. The period of the oscillations is thus 

T = 2n W(LC). 

This is the result obtained by Lord Kelvin in 1857 and confirmed by 
the experiments of Fedderson in 1857. The oscillatory character of the 
discharge had been suspected by Joseph Henry from observations on the 
magnetization of needles placed inside a coil in a discharging circuit. 

In the case of two coupled circuits (L l9 C^), (L 2 , (7 2 ) the mutual induction 
M needs to be taken into consideration and the equations for free oscil- 
lations are ~ 

L& + MQ, + / - 0, 



1-33. The Lagrangian equations of motion. Consider a mechanical 
system consisting of / material points of which a representative one has 
mass m and co-ordinates x, y, z at time /. Using square brackets to denote 
a summation over these material points, we may express d'Alembert's 
principle in the Lagrangian form 

[m (xSx + y8y -f 5 82)] - [XSx + Y8y + ZSz], 

where 8x 9 //, 8z are arbitrary increments of the co-ordinates which are 
compatible with the geometrical conditions limiting the freedom of motion 
of the system. On account of these conditions, the number of degrees of 
freedom is a number N 9 which is less than 3J, and it is advantageous to 
introduce a set of "generalised 11 co-ordinates q l9 g 2 , ... g v which are inde- 
pendent in the sense that any infinitesimal variation 8q s of q s is compatible 
with the geometrical conditions. These conditions may, indeed, be expressed 
in the form 

#=/(?! >?2 <LV> 0> y = 9 (?1>?2> 1\'>0> * = M<?1,?2> ' <7A>0- 

Using the sign S to denote a summation from 1 to N, a prime, to denote 
a partial differentiation with respect to t and a suffix s to denote a partial 
differentiation of x 9 y or z with respect to q s , we have the equations 

x = x' -f Zx s q 89 Sx = I<x 3 8q S9 



where the quantities Q (s) may be called generalised force components 
associated with the co-ordinates q. The first of these equations shows that 
x s is also the partial derivative of x with respect to q s and so if the kinetic 
energy of the system is T 9 where 

2T = [m (x 2 4- y 2 + z 2 )], 
we shall have -_ 

Cl r 



Lagrange's Equations * 31 

where p 8 is the generalised component of momentum. Since 
dx Sx f v . dx r 
Sq^Wr^ " Xr * qs ^~dt =s * r ' 

and -.-- (xx s ) = xx s -f #x g , 

rt* 

we have 

[m (xSx + ySy + 282)] - S8g s , m (^ + yy s + 22,) - m (xx s -f yy, + zz s ) 

i ai . J 



and so Lagrange's principle may be written in the form 

dT\ v-/ir<,^ 

-** * 9 " 



On account of the arbitrariness of the increments 8q s this relation gives 
the Lagrangian equations of motion 

dfST^dT^ 
dt\dq s ) dq s ^ ' 

If there is a potential energy function F, which can be expressed simply* 
in terms of the generalised co-ordinates </, we may write 



and the equations of motion take the simple form 

d i dL\ dL 



The quantity L is called the Lagrangian function. 
Introducing the reciprocal function 

T-x(a dT ] T 

7 -M ? *a$J ' 

, dT _ f . d /ST\ .. dT) /.. ST . dT\ 

we have -- = S , L < ,-.- - S U + - 



Hence we have the energy equation 

T 7 -f F = constant. 

When the functions /, g and h do not contain the time explicitly we 
have on account of Euler's theorem for homogeneous functions 



The Lagrangian equations of motion may be replaced by another set of 
equations for the quantities p and q. For this purpose we introduce the 
Hamiltonian function H defined by 

= - L+ ^p 8 q B - 



32 The Classical Equations 

If we always consider H as a function of the quantities q 8 and p 8 but 
L as a function of q 8 and q a , we have 



Thus * H -=q., m =-^~, 

consequently the equations of motion can be expressed in the Hamiltonian 



^_ 

dt ~~ dp, ' dt ~ dq, * 

Systems of equations whose solutions represent superposed simple 
harmonic vibrations are derived from the Lagrangian equations of motion 
of a dynamical system 

d /dT\ _ dT __ _ SV 
dt \dqj dq s ~ dq s ' 

5= 1,2, ... A T , 

whenever the kinetic energy T, and the potential energy V, can be ex- 
pressed for small displacements and velocities in the forms 

N N 

2T= S 2 a mn q m q n , 

m-l n-1 

N N 

2V = S S c mn?m ? n 

7/1-1 711 

respectively, where the constant coefficients a mn and c mn are such that T 
and F are positive whenever the quantities q m , q m do not all vanish. For 
such a system the equations (A) give the differential equations 

N 

Knrtfn + Cin?) ^ > 
n-1 

m- 1, 2, ... N. 

Multiplying by u m , where u m is a constant to be determined, and 
summing with respect to m, the resulting equation is of type 

v + k*v= 0, (B) 

if the quantities u m are such that for each value of n 

N N 

S u m c mn - fc 2 S w ro a ww - P6 n , say. 

?nl m-l 

The corresponding value of v is then 

N 

v= S 6 n ? n . 

n-l 



Normal Vibrations 33 

Now when the quantities u m are eliminated from the linear homo- 
geneous equations N 

2 (c mn - k 2 a mn ) u m = 0, (C) 

?n = l 

we obtain an algebraic equation of the A 7 th degree for k 2 . With the usual 
method of elimination this equation is expressed by the vanishing of a 
determinant and may be written in the abbreviated form 

I c k 2 n I 
I ^mn K a mn \ u 

To show that the values of k 2 given by this equation are all real and 
positive, we substitute k 2 = h -f ij, u m = v m -f iw m in equation (C). 
Equating the real and imaginary parts, we have 

N N 

2 (c mn - Jui mn ) v m + j 2 a mn w m = 0, 

m 1 m 1 

AT iv 

2 (c mn - Aa mn ) w n -j 2 a mn t? w - 0. 

ra = 1 ?7i 1 

Multiplying these equations by w n and v n respectively, adding and 
summing, we find that 

j 2 a mn (w m w n -f v m v n ) = 0. 

in, n 

The factor multiplying j is a positive quadratic form which vanishes 
only when the quantities v n , w n are all zero, hence we must have j and 
this means that k 2 is necessarily real. That k is necessarily positive is seen 
immediately from the equation 

2 c mn u m u n = k 2 2 a mn u m u n , 

m, n m,n 

which involves two positive quadratic forms. 

If u m (A^), u m (k 2 ) are values of u m corresponding to two different values 
of k we have the equations 

2 (C mn - *!) Um (*i) = 0, 

7/J-l 

N 

S (c wn - k 2 2 a mn ) u m (k,) = 0. 

TO-l 

Multiplying these by w n (A: 2 ), w n (^ 2 ) respectively and subtracting we 
find that (V _ V) s a ^ Wm (Aj) ^ (k j = Q (D) 

m,n 

Denoting the constant 6 W associated with the value k by 6 n (^), we see 
from the last equation that if k 2 = k l9 

S 6 n (*i) ^n (*i) = 0. 
n-1 

On the other hand, 

N 

2 6 n fa) u n (k^ = 2 a mn u m (*J w n (^), 

n 1 m, n 



34 The Classical Equations 

and is an essentially positive quantity which may be taken without loss of 
generality to be unity since the quantities u n (k^ contain undetermined 
constant factors as far as the foregoing analysis is concerned. 

Using the symbol v (k, t) to denote the function v corresponding to a 
definite value of k, we observe that if 

N 

q m = S v(k 8 ,t) A ms , 

8-1 

N 

we must have S b n (k 3 ) \ ms = nm 

s-l 

= 1 n = ra. 
Multiplying by u n (k r ) and summing with respect to n we find that 

*mr = U m (k r ). 

N 

Hence q m = I, u m (k s ) v (k s , t). 

.9-1 

This expresses the solution of our system of differential equations in 
terms of the simple harmonic vibrations determined by the equations of 
type (B). The analysis has been given for the simple case in which the 
roots of the equation for k 2 are all different but extensions of the analysis 
have been given for the case of multiple roots. 

The relation (D) may be regarded as an orthogonal relation in general- 
ised co-ordinates. When 

a mn = 0, mn, a mm = 1, 

the relation takes the simpler form 
.v 

2 u m (k v ) u m (k q ) = p + q. 

m -1 

1-34. An interesting mechanical device for combining automatically 
any number of simple harmonic vibrations has been studied by A. Gar- 
basso*. 

A small table of mass m is supported by four light strings of equal 
length / so that it remains horizontal as it swings like a pendulum. The 
table is attached at various points to n simple pendulums (l s , ra s ), 
s = 1, 2, ... n. Each string is regarded as light and is supposed to oscillate 
in a vertical plane and remain straight as the apparatus oscillates. 

Specifying the configuration of the apparatus by the angular variables 
$o> ^i > n we have in a small oscillation 



V = 

* Vorlesungen uber Speklroskopie, p. 65; Torino Atti, vol. XLIV, p. 223 (1908-9). The case in 
which n = 2 has been studied in connection with acoustics by Barton and Browning, Phil. Mag. 
(6), vol. xxxiv, p. 246 (1917); vol. xxxv, p. 62 (1918); vol. xxxvi, p. 36 (1918) and by C. H. Lees, 
ibid. vol. XLVHI (1924). 



Compound Pendulum 
The equations of motion are 



m s 



8=1 



n \ 

2 m s } = 0, 

-1 / 



35 



(8= 1,2, ...n). 

n 

Writing m a = c s m , S c s = c, the equation for k 2 is in this case 



+ C ) - 
-/ P 

- I k z 



- gc, - gc 2 ... - gc n | =o, 
-Z^ o ... | 

g-l 2 k*... I 







or 



g- 







Expanding the determinant we obtain the equation 



where 



/ (V) = II (g - l,k*). 

.s-1 



Now (1 - l s ) [g - (1 + l s ) k*} = l a (g- I lc*) -l,(g- l,k*), 
consequently the equation for k 2 can be written in the form 

/(*) \(g - kk*) fl + gk 2 j- - r ~f -rJ - g S C8 ' s 1 = 0. 

( L S-l "O ~ 1 8) \ ( J L S^ )J 8-1 ^0 ~ ^S) 

If the mass of each pendulum is so small in comparison with that of the 
table that we may neglect terms of the second order in the quantities c s , 
the equation may be written in the form 

(g - U - 17 S l > } (g - l,k* + ^-) . 

\ .s-1 ^0 ~ 1 J .s-1 \ ^0 ~~ 's/ 

Hence the periods of the normal vibrations are approximately 



- 1,2, ...n). 



3-2 



36 The Classical Equations 

If IQ > 1 8 the period of the 5th pendulum is decreased by attaching it to 
the table. If 1 < l s the period is increased. 

EXAMPLES 

1. A simple pendulum (6, N) is suspended from the bob of another simple pendulum 
(a, M ) whose string is attached to a fixed point. Prove that the equations of motion for 

small oscillations are ^ Tv .. ^ T 

(M + N) a 2 d + Nab<f> + (M + N) gaS = 0, 

Nab'B + Nb*j> + Ngb<f> = 0, 
where B and < are the angles which the strings make with the vertical. 

2. Prove that the coefficient of coupling of the compound pendulum in the last example 
is given by 



_ ___ 

M + N' 

3. Prove that it is not possible for the centre of gravity of the two bobs to remain fixed 
in a simple type of oscillation. 

4. A simple pendulum (/, 'M) is suspended from the bob of a lath pendulum which is 
treated as a rigid body with a moment of inertia different from that of the bob. Find the 
equations of motion and the coefficient of coupling. 

5. A simple pendulum (I, M) is attached to a point P of an elastic lath pendulum which 
is clamped at its lower end and carries a bob of mass N at its upper end. At time t the 
horizontal displacements of M, N and P are y, z and az respectively, a being regarded as 
constant. By adopting the simplifying assumption that a horizontal component force F at 
P gives N the same horizontal acceleration as a force aF acting directly on N, obtain the 
equations of motion 

IMy + Mg (y - az) - 0, INz + lNn*z = Mga (y - az), 
and show that the coefficient of coupling is given by the equation 

a? 



~ Nri* + Mm** 9 
where lm 2 = g. 

[L. C. Jackson/PAtf. Mag. (6), vol. xxxix, p. 294 (1920).] 

ft. Two masses m and m' are attached to friction wheels which roil on two parallel 
horizontal steel bars. A third mass J/, which is also attached to friction wheels which roll 
on a bar midway between the other two, is constrained to lie midway between the other two 
masses by a light rigid bar which passes through holes in swivels fixed on the upper part of 
the masses. The masses m and m' are attached to springs which introduce restoring forces 
proportional to the displacements from certain equilibrium positions. Find the equations of 
motion and the coefficient of coupling. 

This mechanical device has been used to illustrate mechanically the properties of coupled 
electric circuits. [See Sir J. J. Thomson, Electricity and Magnetism, 3rd ed. p. 392 (1904); 
W. S. Franklin, Electrician, p. 556 (1916).] 

7. Two simple pendulums (/ lf Jfj), (/ 2 , M 2 ) hang from a carriage of mass M which, with 
the aid of wheels, can move freely along a horizontal bar. Prove that the equations of motion 

are (M + M 1 + M 2 )x + MJ& + M 2 1 2 2 = 0, 



Quadratic Forms 37 

Hence show that the quantities B 19 9 2 can be regarded as analogous to electric potential 
differences at condensers of capacities M^g and M 2 g, the quantities M l l l gd l and M z l 2 g6 z as 
analogous to electric currents in circuits, the quantities 

, 
) 

as analogous to coefficients of self-induction and [(M l + M 2 + M ) g 2 ]~* as analogous to a 
coefficient of mutual induction. 

[T. R. Lyle, Phil. Mag. (6), vol. xxv, p. 567 (1913).] 

8. A simple pendulum of length Z, when hanging vertically, bisects the horizontal line 
joining the knife edges. When the pendulum oscillates it swings freely until the string comes 
into contact with one of the knife edges and then the bob swings as if it were suspended by 
a string of length h. Assuming that the motion is small and that in a typical quarter swing 

10 + go = for < t< T, 

hd + 00 = for r < t < T, 
prove that the quarter period T is given by the equation 

m cot n (T r) = n tan mr, 
where g = Im? = hri 2 . 

1-35. Some properties of non-negative quadratic forms*. Let 

n t n 

g = 2 g rs x r x s 
i, i 

be a quadratic form of the real variables x l9 ... x n , which is negative for no 
set of values of these variables, then there are n linear forms 



1 
with real coefficients p rg such that 

n / n 

g = S ^ 2 p rs x, 
This identity gives the relation 

n 

g lk = S ^ rt ^ rfc 
i 

which can be regarded as a parametric representation of the coefficients in 
a non-negative form. 

This result may be obtained by first noting that g ss is not negative, for 
g^ is the value of g when x r = 0, r s and x s = 1 . If the coefficients g r9 
are not all zero the coefficients g ss are not all zero, because if they were and 
if, say, 12 <0 a negative value of g could be obtained by choosing x 1 = 1, 
x 2 = T 1, # 3 = ... o: n = 0. 

We may, then, without loss of generality assume that there is at least 
one coefficient g u of the set g 88 which is positive. 

* L. Fejer, Math. Zeits. Bd. i, S. 70 (1918). 



38 The Classical Equations 

Writing p n p 18 = g ls s = 1, 2, ... n, 

n 
2j = S p ls X s , 

8-1 

<7<" = g - ,, 

it is easily seen that the quadratic form g (l} does not depend on x v g (l} is 
moreover non-negative because if it were negative for any set of values 
of # 2 , # 3 , ... x n we could obtain a negative value of g by choosing x l so that 
2l = 0. ' 

Since g (l) is non-negative it either vanishes identically or the coefficient 
of at least one of the quantities # 2 2 , x 3 2 , ... x n 2 in f/ {1) must be positive. 
Let us suppose that gr 22 (1) is positive and write 

(1> r = 2, 3,...tt/ 



0(2) _ yd) _ z 2 2. 

Continuing this process it is found that g = 2^+ z 2 2 4- . . .z n 2 , where the 
linear forms ^ , z 2 , . . . z n are not all zero ; it is also found that none of the 
quantities g n , <7 22 (1) , fe* 2 *, ... are negative and that all these quantities, 
except the first, are ratios of leading diagonal minors in the determinant 

| 9rs | , 

and are not all zero. 

n, n 

Now let h = h tk x t x k 

i, i 

be a second non-negative form, and let 

n 
h ik = S Jat^fc 

be its parametric representation, then* 

n, n n,n n 

2 g tk h tk = 2 ( S 

1,1 1,1 \<r-l 

If 0> 2/i, ^2 2/n are arbitrary real quantities, 

2 (x, - 6y s y 

is never negative. Regarding this as a quadratic expression in it is 
readily seen that the quadratic form 

A~S*,*y s -(*.y.) 
i i i 

is non-negative. This result, which was known to Cauchy and Bessel, is 
frequently called Schwarz's inequality as Schwarz obtained a similar 
inequality for integrals. 

* L. Fejr, Math. Zeits. Bd. i, S. 70 (1918). 



Hermitian Forms 39 

Using this particular form of h in Fejer's inequality we obtain the result 
that 

n, n n n 

Xg rs yry,<X9rrXy*. 

1,1 11 

For further properties of quadratic forms the reader is referred to Brom- 
wich's tract, Quadratic Forms, Cambridge (1906), to Bocher's Algebra, 
Macmillan and Co. (1907), and to Dickson's Modern Algebraic Theories, 
Sanborn and Co., Chicago (1926). 

1-36. Hermitian forms. Let z denote the complex quantity conjugate 
to a complex quantity z and let the complex coefficients c rs be such that 



n,n 

the bilinear form 2 c rs z r z s 

i, i 

is then Hermitian, If c lm = a lm + ib lm ,z m = r m e lQm , where a lm ,b lm , r m , 9 m 
are real quantities, we have 

a lm ~ a ml> bi m = O m i, 

and the Hermitian form can be expressed as a quadratic form 

n,n 

2 p tm rir m , 
i, i 

where p lm = a lm cos (0 t - 6J + b lm sin (0 t - m ) = p ml . 

The positive definite Hermitian forms which are positive whenever at 
least one of the quantities z l , z 2 , . . . z n is different from zero are of special 
interest. In this case the associated quadratic form is positive for all non- 
vanishing sets of values of r l , r 2 , . . . r n and for all values of 1 , 6 2l ... 6 n . 
An important property of a Hermitian form is that the associated secular 
equation 

| c rs - A8 r . | = 

8 r3 = 1 r = s 
- r + s 

has only real roots. When the form is positive these roots are all positive. 
The proof of this theorem may be based upon analysis very similar to that 
given in 1-33. 

EXAMPLES 
1. If F(t) ^ for - TT ^ * ^ TT and 



= 2 c e M , 

\> =s ~ 00 

n, n 
#n= 2 Cl-mZlZm n 1,2,3..., 

then # n ^ 0. This has been shown by Carathodory and Toeplitz to be a necessary and 
sufficient condition that F(t) > 0. See Rend. Palermo, t. xxxn, pp. 191, 193 (1911). 



40 The Classical Equations 

2. If /(*)= f e ite o> (x)dx, 

J oo 

where o> (a;) = to ( #) 

n, n _ 

ft nd #= 2 a) (a?j - * m ) f z f m , 

Mathias has shown that when f(t) ^ we have H n ^ for any choice of real parameters 
x l , x 2 , . . . x n and of the complex numbers f, , f, , . . . f n . See Jlf ath. Zeite. Bd. xvi, p. 103 ( 1923). 
The analysis depends upon Fourier's inversion formula which is studied in 3-12 and it 
appears that, with suitable restrictions on the function <*>(x), the inequality H n ^> is the 
necessary and sufficient condition that f(t) ^ 0. Mathias gives two methods of choosing a 
function a>(x) which will make H n ^ 0. The correctness of these should be verified by the 
reader. 

(1) If the functions x( x )> x( x ) are such that when x has any real value x( x ) an( l X ( x ) are 
conjugate complex quantities, the function 

/> 
X(a+ %)\(a - x)da 
-00 

will make H n ^ 0. 

(2) If the positive constants \ v and the functions x v ( x ) are such that 



is a function of x y> say o> (x y) y then this function o> (x) will make H n ^ 0. 

1-41. Forced oscillations. When a particle, which is normally free to 
oscillate with simple harmonic motion about a position of equilibrium, is 
acted upon by a periodic force varying with the time like sin (pt), the 
equation of motion takes the form 

x 4- k 2 x = A sin pt. 

Writing x = z -f- C sin pt, where G is a constant to be determined, we 
find that if we choose C so that 

(k*-p*)C = A, ...... (A) 

the equation for z takes the form 

z -f k*z = 0. 

The motion thus consists of a free oscillation superposed on an oscilla- 
tion with the same period as the force. In other words the motion is partly 
original and partly imitation. It should be noticed, however, that if p* > k 2 
the imitation is not perfect because there is a difference in phase. The 
difference between the case p* > k 2 and the case p 2 < k 2 is beautifully 
illustrated by giving a simple harmonic motion to the point of suspension 
of a pendulum. 

When p 2 == k 2 the quantity C is no longer determined by equation (A) and 
the solution is best obtained by the method of integrating factors which 
may be applied to the general equation 

x + k 2 x = F (t). 



Forced Oscillations 41 

Multiplying successively by the integrating factors cos kt and sin kt and 

integrating, we find that if x = c, x = u when t = T, we have 

ft 
x cos kt -f fc sin kt = u cos &T -f &c sin kr + \ F (s) cos ks . d#, 

r< 
# sin kt kx cos kt u sin &r kc cos &r + \ F (s) sin &s . ds, 

rt 
kx = u sin A; (t r) -f &c cos k (t r) -f F (s) sin &($ *) ds, 

ct 
x = u cos A: (t r) &c sin k (t r) -f I F (s) cos k (t s) ds, 

J T 

the function F (s) being supposed to be integrable over the range T to t. 

In particular, if the particle starts from rest at the time t we have at 
any later time t 

kx = IF (s) sin k (t s) ds, 

rt 
x = IF (s) cos k (t s) ds. 

J r 

When F (s) = A sin ks and r = we find that 
2fcr = t cos itf &" 1 sin kt, 

and the oscillations in the value of x increase in magnitude as t increases. 
This is a simple case of resonance, a phenomenon which is of considerable 
importance in acoustics. In engineering one important result of resonance 
is the whirling of a shaft which occurs when the rate of rotation has a 
critical value corresponding to one of the natural frequencies of lateral 
vibration of the shaft. For a useful discussion of vibration problems in 
engineering the reader is referred to a recent book on the subject by 
S. Timoshenko, D. Van Nostrand Co., New York (1928). 

By choosing the unit of time so that k = 1 the mathematical theory 
may be illustrated geometrically with the aid of the curve whose radius 
of curvature, p, is given by the equation p = a sin OHJJ, where ^ is the angle 
which the tangent makes with a fixed line. Using p now to denote the 
length of the perpendicular from the origin to the tangent, we have the 
equation ^ 

...... (B) 



The quantity p thus represents a solution of the differential equation, 
and by suitably choosing the position of the origin the arbitrary constants 

in the solution can be given any assigned real values. In this connection 

d*D 
it should be noticed that ~j has a simple geometrical meaning (Fig. 7). 

When co = 1 the equation (B) is that of a cycloid, while epicycloids and 
hypocycloids are obtained by making a) different from unity. The intrinsic 
equation of these curves is in fact 

4 (a -f b) b . aJj 

Q V ' am ..... T ___ 

o - OIJ.JL rt , , 

a a + 2b 



42 The Classical Equations 

where a is the radius of the fixed circle and b the radius of the rolling circle 
which contains the generating point. 

Epicycloids 6 > 0. 

Pericycloids and hypocycloids 6 < 0. 





Fig. 8. 

1-42. The effect of a transient force in producing forced oscillations 
is best studied by putting r = oo and assuming that c and u are both 
zero, then 

Too 

kx = F (t cr) sin ka . da, 

Jo 



f 00 
x = F (t a) cos ka . da. 

' 



Let us consider first of all the case when 



In this case F' (t) is discontinuous at time t = in a way such that 

F' (- 0) - F' (-f 0) - 2h. 

The solution which is obtained by supposing that x, x and x are con 
tinuous at time t = is 



ft Too 

lex = e-w-*> sin kada -f 

.'0 J 



2 A sin let 
k*+h* 



_ ht 



. _ 2h cos kt he~ ht 

== ~v"o ; ~ o ~ 



i > 



It will be noticed that x is discontinuous at t = 0. 



Residual Oscillation 43 

It is clear from these equations that x increases with t until t reaches 
the first positive root of the transcendental equation 

2 cos kt = e~ ht . 
The corresponding value of x is then 

2 (h sin kt -f k cos kt) 

As t increases beyond the critical value x begins to oscillate in value, 
and as t -> oo there is an undamped residual oscillation given by 

_ 2h sin kt 

A general formula for the displacement x at time t in the residual 
oscillation produced by a transient force may be obtained by putting 
t = oo in the upper limit of the integral (t being retained in the integrand) *. 
This gives 

kx = I F (s) sin k (t s) ds. 

J -00 

In particular, if 

,, , x [ b 7 sin bs 

F (s) = cos ms . dm = - , 

Jo s 

we have 

/*QO f? Q r ^ ft ^ 

kx sin kt cos ks sin bs cos kt sin ks sin 65 ; 

.' -oo S J _oo S 

the second integral vanishes arid we may write 

TOO ^ 

2&# = sin kt [sin (ft -f k) s -f sin (6 k) s] 

J -oo S 

= 277 sin & if 6 > & > 
' = if < b < k 

= TT sin kt if b = & > 0. 

There is thus a residual oscillation only when b > k. 

EXAMPLES 

/& 

1. If F (s) = I , cos m* . dm, 

there is a residual oscillation only when k lies within the range a < &< 6. Extend this result 
by considering cases when 

F ( 8 ) = / cos ms . <f> (m) dm, F (s) = / sin ms . </> (m) dm, 
J a J a 

^ (m) being a suitable arbitrary function. 

2. Determine the residual oscillation in the case when 

F (s) - (c 2 -f s 2 )- 1 . 
* Cf. H. Lamb, Dynamical Theory of Sound, p. 19. 



44 The Classical Equations 

3. If F (s) = se~ h I s I , where h > and # is chosen to be a solution of the differential 
equation and the supplementary conditions x 0, x 0, when t = oo , x and x continuous 
at t 0, the residual oscillation is given by 

4th cos kt 

X ^~(k*~+h*) 2 ' 
If k 2 > h 2 there is a negative value of t for which r 0, but if k 2 < h 2 there is no such value. 

4. If .c + k*x = x<*(t)> 

where o>(/) is zero when t~ oo and is bounded for other real values of t, the solution for which 
x and b are initially zero can be regarded as the residual oscillation of a simple pendulum 
disturbed by a transient force. 

5. If < a 2 < A (t) < 6 2 , the differential equation 

.- + A(t)x = T) (C) 

is satisfied only by oscillating functions. Prove that the interval between two consecutive 
roots of the equation x lies between rr/a and TT/&. 

[Let y be a solution oy 4- 6 2 y = which is positive in the interval r < t < r 2 , then 



Let us now suppose that it is possible for x to be of one sign (positive, say) in the interval 
T, ^ t < T 2 and zero at the ends of the interval. We are then led to a contradiction because 
the suppositions make .r positive near ^ and negative near r 2 , they thus make the left-hand 
side negative (or zero) and the right-hand side positive. Hence the interval between two 
consecutive roots of x must be greater than any range in which y is positive, that is, greater 
than y/b. In a similar way it may be shown that if z is a solution of z -f a 2 z = the interval 
between two consecutive roots of the equation z is greater than any range in which x ^ 0.] 

This is one of the many interesting theorems relating to the oscillating functions which 
satisfy an equation of type (C). For further developments the reader is referred to Bocher's 
book, Lemons sur les methodes de Sturm, Gauthier-Villars, Paris (1917) and his article, 
" Boundary problems in one dimension," Fifth International Congress of Mathematicians, 
Proceedings, vol. I, p. 163, Cambridge (1912). 

6. Prove that x 2 -f x 2 remains bounded as t -> x but may not have a definite limiting 
value, x being any solution of (C). [M. Fatou, Compte* Rendus, t. 189, p. 967 (1929).] 
The generality of this result has recently been questioned. See Note I, Appendix. 

1-43. Motion with a resistance proportional to the velocity. Let us first 
of all discuss the motion of a raindrop or solid particle which falls so slowly 
that the resistance to its motion through the air varies as the first power 
of the velocity. This is called Stokes' law of resistance; it will be given in 
a precise form in the section dealing with the motion of a sphere through 
a viscous fluid ; for the present we shall use simply an unknown constant 
coefficient k and shall write the equation of motion in the form 

mdv/dt = m'g kv, (A) 

where m is the apparent mass of the body when it moves in air, m' is the 
reduced mass when the buoyancy of the air is taken into consideration and 
g is the acceleration of gravity. The solution of this equation is 

_kt 

kv = m'g + Be m , 



Resisted Motion 45 

where -B is a constant depending on the initial conditions. If v = when 
t = we have tct 

kv = m'g(\ - e '*). 
To find the distance the body must fall to acquire a specified velocity 

v we write v = x , , , , 7 

mvdv/dx = mg kv, 



/ i 

'gr log -- 

y & Vra - kv 

If T 7 denote the terminal velocity the equation may be written in the 
form v 

m'gx = ra F 2 log -=%- -- m Vv. 

Let us now consider the case in which the particle moves in a fluctuating 
vertical current of air. Let /' (t) be the upward velocity of the air at time 
t, v the velocity of the particle relative to the ground and u = v + / x (t) the 
relative velocity. On the supposition that the resistance is proportional to 
the relative velocity the equation of motion is 

mdv/dt = m'g ku --- m'g kv ~ kf (t). 

If v = when t = the solution is 



The last integral may be written in the form 



/'(t-r)e "'dr, 

/ 

which is very useful for a study of its behaviour when t is very large. 
The distance traversed in time / is given by the equation 



the constant in/ (t) being chosen so that/ (0) = 0. In particular if 

~~~k~ ' p' 

i ki 

clc 

we have v = - - ,-* [ke m m P sin pt k cos pi] , 

m 2 p 2 + k 2L * r r j 

ck * 

r 7 ^ vn~\ 



As t -> oo we are left with a simple harmonic oscillation which is not in 
the same phase as the air current. 

It should be emphasised that this law of resistance is of very limited 
application as there is only a small range of velocities and radius of particle 



46 The Classical Equations 

for which Stokes' law is applicable. It should be mentioned that the product 
of the radius and the velocity must have a value lying in a certain range 
if the law is to be valid. 

An equation similar to (A) may be used to describe the course of a 
unimolecular chemical reaction in which only one substance is being trans- 
formed. If the initial concentration of the substance is a and at time t 
altogether x gram-molecules of the substance have been transformed, the 
concentration is then a x and the law of mass action gives 

dx . 



the coefficient k being the rate of transformation of unit mass of the substance. 
Simultaneous equations involving only the first derivatives of the 
variables and linear combinations of these variables occur in the theory 
of consecutive unimolecular chemical reactions. If at the end of time t the 
concentrations of the substances A, B and C are x, y and z respectively 
and the reactions are represented symbolically by the equations 

A-+B, B-+C, 
the equations governing the reactions are 

dx/dt = &!#, dy/dt = ^x k 2 y, dz/dt = k 2 y. 

A more general system of linear equations of this type occurs in the 
theory of radio-active transformations. Let P , P ly ... P n represent the 
amounts of the substances A , A 1 , ... A n present at time t, then the law of 
mass action gives , p 

_- A P 
dt ~~~ ** OJ 



~^ ^n-lPn-l 

where the coefficients A s are constants. In my book on differential equations 
this system of equations is golved by the method of integrating factors. 
This method is elementary but there is another method* which, though 
more recondite, is more convenient to use. 

Let us write p s (x) = f V*'P 8 (t) dt, (B) 

J Q 
Too fjp 

then e-* ~ 8 dt = - P 8 (0) + xp s (x), 

Jo dt 

* H. Bateman, Proc. Camb. Phil. Soc. vol. xv, p. 423 (1910). 



Reduction to Algebraic Equations 47 

and so the system of differential equations gives rise to the system of 
linear algebraic equations 

xp, (x) - P (0) = - A^ (x), 

xp l (x) - P l (0) = AO^ O (x) ~ \p l (x), 

xp 2 (x) - P 2 (0) = Aj^ (x) - X 2 p 2 (x), 



Xp n (X) - P n (0) = V,1>-1 (X) - 

from which the functions p Q (x), p t (x), ... p n (x) may at once be derived. 

If P l (0) = P 2 (0) = ... = P tt (0) = 0, i.e. if there is only one substance 
initially, and if P (0) = Q, we have 

p (x) = x TV Pl (x} = ~(*TAo)7aTXj ' 



To derive P s (t) from j9 s (x) we simply express p s (x) in partial fractions 

p te) = - C ____ h ... - - 

X ~T" AQ X ~\- Ag 

The corresponding function P s (t) is then given by 



for this is evidently of the correct form and the solution of the system of 
differential equations is unique. The uniqueness of the function P s (t) 
corresponding to a given function p s (x) can also be inferred from Lerch's 
theorem which will be proved in 6-29. 

If some of the quantities A w are equal there may be terms of type 



in the representation of p s (x) in partial fractions. In this case the corre- 
sponding term in P s (t) is n _ A t 

^m,* t < e "t . 

Such a case arises in the discussion of a system of linear differential 
equations occurring in the theory of probability*. 

1-44. The equation of damped vibrations. A mechanical system with 
one degree of freedom may be represented at time t by a single point P 
which moves along the x-axis and has a position specified at this instant 
by the co-ordinate x. This point P, which may be called the image of the 
system, may in some cases be a special point of the system, provided that 
the path of such a point is to a sufficient approximation rectilinear. The 

* H. Bateman, Differential Equations, p. 45. 



48 The Classical Equations 

mechanical system may also in special cases be just one part of a larger 
system ; it may, for instance, be one element of a string or vibrating body 
on which attention is focussed. To obtain a simple picture of our system 
and to fix ideas we shall suppose that P is the centre of mass of a pendulum 
which swings in a resisting medium. 

The motion of the point P is then similar to that of a particle acted upon 
by forces which depend in value on t, x, x and possibly higher derivatives. 
For simplicity we shall consider the case in which the force F is a linear 

function of x and x, /. m , m 07 u . . 

F = / (t) h (t) x 2k (t) x. 

In the case when h (t) and k (t) are constants the equation of motion 

takes the simple form rt7 . 9 /jt . , A v 

* x + 2kx + n 2 x = f (t). (A) 

The motion of the particle is in this case retarded by a frictional force 
proportional to the velocity. If it were not for this resistance the free 
motion of the particle would be a simple harmonic vibration of frequency 
ft/277. The effect of the resistance when n 2 > k 2 is to reduce the free motion 
to a damped oscillation of type 

x = Ae~ kt sin (pt + e), (B) 

where A and are arbitrary constants and 

pz = n 2 - k 2 . 

The period of this damped oscillation may be defined as the interval 
between successive instants at which x is a maximum and is %TT/P. One 
effect of the resistance, then, is to lengthen the period of free oscillations. 
It may be noted that the interval between successive instants at which 
x = is 2-rr/p. The time range < t < oo may, then, be divided up into 
intervals of this length. The sign of x changes as t passes from one interval 
to the next and so the point P does in fact oscillate. Points P and P' of 
two intervals in which x has the same sign may be said to correspond if 
their associated times t, t', are connected by the relation 

pt' = pt + 2m7r, 
where m is an integer. We then have 

x' = xe~ k(t '~ t} = xe~ 2kmir l p . 

The positive constant k is seen, then, to determine the rate of decay 
of the oscillations. 

When n 2 = k 2 the free motion is given by 

x = (A + Bt) e~ kt , 

where A and J5 are arbitrary constants. In this case x vanishes at a time 
t given by B = k (A -f- Hi), thus | x \ increases to a maximum value and 
then decreases rapidly to zero. The motion of a dead-beat galvanometer 
needle may be represented by an equation of type (A) with n 2 = k 2 . 



Damped Vibrations 49 

When k 2 > n 2 the free motion is of type 

x = Ae~ ut -f fie-"', 
where ^ and v are the roots of the equation 

2 2 ~ 2A'Z + 7l 2 - 0. 

In this ease the general value of x is obtained by the addition of two 
terms each of which represents a simple subsidence, the logarithmic de- 
crement of which is u for the first and v for the second. The time r which 
is needed for the value of x in one of these subsidences to fall to half value 

is given by the equation ^ __ 

2 e UT - 

In the case of the damped oscillation (B) the quantity Ae~ ki can be 
regarded as the amplitude at time t. In an interval of time r this diminishes 
in the ratio r: 1, where r = e kr . 

Putting T = 1, we have k -- log r, whence the name logarithmic decre- 
ment usually given to k. Instead of considering the logarithmic decrement 
per unit time we may, in the case of a damped vibration, consider the 
logarithmic decrement per period or per half-period*. It is knjp for the 
half -period. 

When / (t) C sin mt, where C and m are constants, the solution of 
the differential equation (A) is composed of a particular integral of type 

~ (n 2 m 2 ) sin nit 2km cos mt 

X ~ C . 0\0~ , A 1 9 "T" ...... (I) 

(n* m 2 ) 2 -f 4:k~m 2 v ' 

and a complementary function of type (B). The particular integral is 
obtained most conveniently by the symbolical method in which we write 
D = djdt and make use of the fact that D 2 f m 2 /. The operator D is 
treated as an algebraic quantity in some of the steps 



____ 

-f- 2A-/> H- n 2 ~ ^ 2 - m 2 -f 2k 
2 -m 2 -- 2kD 



_ __ 2 ^ ( '' 

If .r = 0, a; = when ^ = a the unknown constants in the complementary 
function may be determined and we find that 

- r)f(r)dr. ...... (C) 

This result may be obtained directly from the differential equation by 
using the integrating factors e kt sin pt and e u cos pt. 
We thus obtain the equations 

px (*< sin pt + (x + kx) *' cos p< = | e* r cos pr ./ (r) dr> 



cos 



fc-r) ^' sin p< = e kr sin pr ./ (T) 



* E. H. Barton and E. M. Browning, Phil. Mag. (6), vol. XLVII, p. 495 (1924). 
B 4 



50 The Classical Equations 

from which (C) is immediately derived. We also obtain the formula 

X + kx - [ e~ k(t - T) COSp (t - r)f(r) dr. 

introducing an angle c defined by the equation 

2km 

tan e= 2 2 
n 2 m 2 

the particular integral (I) may be expressed in the form 

x ~ A siri (mi e), 
where the amplitude A is given by the formula 

A 2 [(n 2 ~ m 2 ) 2 + 4k 2 m 2 ] - C 2 . 

N 

This is the forced oscillation which remains behind when the time t is so 
large that the free oscillations have died down. The amplitude A is a 
maximum when m is such that m 2 ^ n 2 2k 2 =- p 2 k 2 . We then have 

^4max= C/2kp. 

Writing A = a/l max it is easily seen that when m is nearly equal to n 
and k/n is so small that its square can be neglected we have the approxi- 
mate formula* 7 , , ,, . i 

k = a I n m \ (1 a 2 ) *. 

This formula has been used to determine the damping of forced oscillations 
of a steel piano wire. 

It should be noticed that a differential equation of type (A) may be 
obtained from the pair of equations 

u -f ku -f nv 0, 
v -f kp nu = 0, 

where k and n are constants. These represent the equations of horizontal 
motion of a particle under the influence of the deflecting force of the earth's 
rotation and a frictional force proportional to the velocity. These equations 

k uu -f- vv -f k (u 2 -f v 2 ) = 0, 

u 2 + v 2 = q 2 e- 2kt , 

hence k is the logarithmic decrement for the velocity. 

The equation of damped vibrations has some interesting applications 
in seismology and in fact in any experimental work in which the motion 
of the arms of a balance is recorded mechanically. 

The motion of a horizontal or vertical seismograph subjected to dis- 
placements of the ground in a given direction, say x f (t), can be repre- 
sented by an equation of form 

6 + 2kti + n 2 + x/l = 0, 

* Florence M. Chambers, Phil Mag. (6), vol. XLVIII, p. 636 (1924). 



Instrumental Records 51 

where 6 is the deviation of the instrument, k a constant which depends 
upon the type of damping and I the reduced pendulum length*. 

The motion of a dead-beat galvanometer, coupled with the seismograph, 
is governed by an equation of type 

$ + 2m(j> -f mty + h9 = 0, 

where </> is the angle of deviation of the galvanometer and h and m are 
constants of the instrument. 

To get rid as soon as possible of the natural oscillations of the pendulum, 
introduced by the initial circumstances, it is advisable to augment the 
damping of the instrument, driving it if possible to the limit of a periodicity 
and making it dead beat. By doing this a more truthful record of the move- 
ment of the ground is obtained. 

When n 2 = k 2 the solution of the equation of forced motion 

x -f 2kx + k*x - / (t) 
is a= * (t- T)e- k(i - r) f(T)dr^ (A + Bt)e~ u . 

When t is large the second term is negligible and the lower limit of the 
integral may, to a close approximation, be replaced by 0, oo, or any 
other instant from which the value of/ (t) is known. 

When k is large the second term is negligible even when t has moderate 
values and if, for such values of t, f (t) is represented over a certain range 
with considerable accuracy by C sin mt, the value of x is given approxi- 
mately by the formula 

f(t) _ Csinmt 



_ 
X ~ D* + 2kJ) + T 2 " k 2 - m 2 + 2kD 



...(I') 



When k is large in comparison with m a good approximation is given by 

k*x =- C sin mt =/(/)> 

and the factor of proportionality k 2 is independent of m, consequently, 
if a number of terms were required to give a good representation of / (t) 
within a desired range of values of t, the record of the instrument would 
still give a faithful representation, on a certain definite scale, of the 
variation of the force. 

When k and an are of the same order of magnitude this is no longer true, 
consequently, if the "high harmonics" occur to a marked degree in the 
representation of/ (t) by a series of sine functions, the record of the instru- 
ment may not be a true picture of the forcef. 

* B. Galitzm, "The principles of instrumental seismology," Fifth International Congress 
of Mathematicians, Proceedings, vol. I, p. 109 (Cambridge, 1912). 

f If m = &/10 the solution of the differential equation is approximately k 2 x -99 sin (mt - c), 
where c ia the circular measure of an angle of about 16 59'. When m = k/5 the solution is approxi- 
mately k z x 96 sin (mt c), where e is the circular measure of an angle of about 31 47'. 



52 The Classical Equations 

When n ^ k the formula (I') shows that if k is large in comparison with 
both m and n the solution is given approximately by the formula 

2kmx = ~ C cos mt 
which may be written in the form 



This result may be obtained directly from the differential equation by 
neglecting the terms x and n 2 x in comparison with 2kx. In this case the 
velocity x gives a faithful record of the force on a certain scale. 

Finally, if n 2 is large in comparison with k and ra, formula (I) gives 



and the instrument gives a faithful record of the force when the natural 
vibrations have died down. 

1-45. The dissipation function. The equation of damped vibrations 

mx + kx + {JLX = f (t) 
may be written in the form of a Lagrangian equation of motion 

d dT\ dF 3V 



where T - \mx\ F - \kx\ V - \^x\ 

Regarding m as the mass of a particle whose displacement at time t is 
x, T may be regarded as the kinetic energy, V as the potential energy and 
F as the dissipation function introduced by the late Lord Rayleigh*. The. 
function F is defined for a system containing a number of particles by an 
equation of type p = ^ (/ ^ 2 + ^, + ^ 

where k x , k y , k z are the coefficients of friction, parallel to the axes, for the 
particle x,y,z. Transforming to general co-ordinates #1 , </ 2 > <7a > . . . q n we may 

write 



where the coefficients [rs], (rs), {rs} are of such a nature that the quadratic 
forms T, F, V are essentially positive, or rather, never negative. These 
coefficients are generally functions of the co-ordinates q l9 ... q n , but if we 
are interested only in small oscillations we may regard q l , ... q n , q l9 ... q n 
as small quantities and in the expansions of the coefficients in ascending 
powers of q l9 ... q n it will be necessary only to retain the constant terms 
if we agree to neglect terms of the third and higher orders in (ft, ... q n , 
4i> ? 

* Proc. London Math..Soc. (1), vol. iv, p. 357 (1873). 



Reciprocal Relations 53 

The generalised Lagrangian equations of motion are now 



- 

dt\dq 

where Q m is the generalised force associated with the co-ordinate q m . Since 
T is supposed to be approximately independent of the quantities q lt 
q 2 , ... q n the second term may be omitted. 

Using rs as an abbreviation for the quadratic operator 



the equations of motion assume the linear form 

H& + T2gr 2 + ... = Q 19 
2l ?1 + 22g 2 + ... = Q 2 . ...... (E) 

Since [rs] = [sr], (rs) = (sr), (rs} = {sr}, it follows that 7s = sr. 

1-48. Rayleigh's reciprocal theorem. Let a periodic force Q s equal to 
A s cos pt act on our mechanical system and produce a forced vibration of 



= KA 8 cos (pi - e), 

where X is the coefficient of amplitude and e the retardation of phase. The 
reciprocal theorem asserts that if the system be acted on by the force 
Q r = A cos pi, the corresponding forced vibration for the co-ordinate </ s , 

will be v . . 

q s = KA r cos (pt ). 

Let D denote the determinant 

IT 12 13 .. 

21 22 23 .. 

31 32 33 .. 



and let rs denote its partial derivative with respect to the constituent rs 
when no recognition is made of the relation rs = sr and when all the 
constituents are treated as algebraic quantities. This means that rs is the 
cofactor of rs in the determinant operator D. 

Solving the equations (E) like a set of linear algebraic equations on 
the assumption that D ^ 0, we obtain the relations 

f ...nlQ n , 
...n2Q n , 



From a property of determinants we may conclude that since rs = sr 
we have also rs = sr . Thus the component displacement q r due to a force 
<?. is given by - 



54 The Classical Equations 

Similarly, the component displacement q s due to a force Q r is given by 

Dq 8 = srQ r . 
Distinguishing the second case by adash affixed to the various quantities, 



where the coefficients A s , A/ may without loss of generality be supposed 
to be real. If they were complex but had a real ratio they could be made 
real by changing the initial time from which t is measured. 
Expressing the solution in the form 

T W S7* 

A " 5 pu,t n ' A C lpt 

(J r ^1 8 j~j V , y s si r jpG , 

and defining the forced vibration as the particular integral obtained by 
replacing d/dt in each of the operators by ip, we obtain the relation 

A/q r = A s q; 
which gives reciprocal relations for both amplitude and phase. 

In the statical case the quantities [rs], (rs), are all zero and D, rs, rs 
are simply constants. Rayleigh then gives two additional theorems 
corresponding to those already considered in 2. 

(2) Suppose that only two forces Q l3 Q 2 act, then 



If q l = we have ^ ^ 

2 = [12 2 - 



From this we conclude that if q 2 is given an assigned value a it requires 
the same force to keep q^ = as would be required if the force Q 2 is to keep 
q 2 = when q l has the assigned value a. 

(3) Suppose, first, that Q 1 == 0, then the equations (F) give 

?1 : ?8 =21:22. 

Secondly, suppose q 2 = 0, then 

02:0!= -12:22. 

Thus, when Q 2 acts alone, the ratio of the displacements q l9 q 2 is Q^IQu 
where Q 11 Q% are the forces necessary to keep q 2 0. 

1-47. Fundamental equations of electric circuit theory. A system of 
equations analogous to the system (E) occurs in the theory of electric 
circuits. This theory may be based on KirchhofTs laws. 

(1) The total impressed electromotive force (E.M.P.) taken around any 
closed circuit in a network is equal to the drop of electric potential ex- 
pressed as the sum of three parts due respectively to resistance, induction 
and capacity. 



Electric Circuits 55 

Thus, if we consider an elementary circuit consisting of a resistance 
element of resistance jR, an inductance element of self-induction (or in- 
ductance) L and a capacity element of capacity (capacitance) (7, all in 
series, and suppose that an E.M.F. of amount E is applied to the circuit, 
Kirchhoff's first law states that at any instant of time 

RI + L Tt + 9 C = E > W 

where / is the current in the circuit and Q = \Idt. 

The fall of potential due to resistance is in fact represented by KI 
where R is the resistance of the circuit (Ohm's law), the drop due to in- 
ductance is Ldl/dt and the drop across the condenser is Q/C. 

There is an associated energy equation 



in which RI represents the rate at which electrical energy is being converted 
into heat, while the second and third terms represent rates of increase of 
magnetic energy and electrical energy respectively. The right-hand side 
represents the rate at which the impressed E.M.F. is delivering energy to 
the circuit, while the left-hand side is the rate at which energy is being 
absorbed by the circuit. The inductance element and the condenser may 
be regarded as devices for storing energy, while the resistance is responsible 
for a dissipation of energy since the energy converted into Joule's heat is 
eventually lost by conduction and radiation of heat or by conduction and 
convection if the circuit is in a moving medium. 

If we regard Q as a generalised co-ordinate, we may obtain the equation 
(I) by writing y = ^ f _ ^ V = Q2/2C> 

dF dV _, 
. - . f .- = E. 

dQ vQ 

(2) In the case of a network the sum of the currents entering any 
branch point in the network is always zero. 

If we consider a general form of network possessing n independent 
circuits, Kirchhoff's second law leads to the system of equations 

-E r , (II) 



(r= 1,2, ...n), 

where E r is the E.M.F. applied to the rth circuit, L ss , R 9SJ C KS denote the 
total inductance, resistance and capacitance in series in the circuit s, 
while L rs , R rs , C rs denote the corresponding mutual elements between 
circuits r and s. We have written F rs for the reciprocal of C rs and Q s for 

I/, eft, where I s is the current in the 5th circuit or mesh. 



56 The Classical Equations 

An elaborate study of these equations in connection with the modern 
applications to electrotechnics has been made by J. R. Carson*. 

In discussing complicated systems of resistances, inductances and 
capacities it will be convenient to use the symbol (LRC) for an inductance 
L, a resistance R and a capacity C in series. If a transmission line running 
between two terminals T and T' divides into two branches, one of which 
contains (LRC) and the other (L'R'C'}, and if the two branches subse- 
quently reunite before the terminal T' is reached, the arrangement will be 
represented symbolically by the scheme 



In electrotechnics a mechanical system with a period constant n and 
a damping constant k is frequently used as the medium between the 
quantity to be studied (which actuates the oscillograph) and the record. 
The oscillograph is usually critically damped (k = n) so as to give a faithful 
record over a limited range of frequencies but even then the usefulness of 
the instrument is very limited as the range for accurate results is given 
roughly by the inequality 10m < n. 

A method of increasing the working range of such an oscillograph has 
been devised recently by Wynn- Williams f. If an E.M.F. of amount E acts 
between two terminals T and T' and the aim is to determine the variation 
of E, the usual plan is to place the oscillograph in line with T and T' so 
that T, and T' are in series. In our notation the arrangement is 

T - - T. 
Instead of this Wynn- Williams proposes the following scheme 



in which L, R and C are chosen so that 2k = JR/L, n 2 = l/CL and L 19 K 19 C 1 
are chosen so that L l , C\ are small and R 1 is such that there is a relation 
(k 4- ^) 2 - n 2 + n^ 9 where 2k, - J?,//,, nf = l/LC^ 

Putting L) = and writing q for the current flowing between T and 
T' , x for the reading of the oscillograph, F for the back E.M.F. of the system 
(LRC), we have 



Therefore x = -~ ; 



c\ 

[D 2 + 2kD + n 2 ] x = F. 
E 



* Klcctnc circuit theory and operational calculus (McGraw Hill, 1926). 
f Phil. Mag. vol. L, p. 1 (1925). 



Cauchy's Method 57 

Hence when the reading of the oscillograph is used to determine E the 
oscillograph behaves as if its period constant were (n 2 -f n^ instead of 
n and its damping constant k -f 1c l instead of k. By choosing n^ = 24n 2 , 
& x = 4k = 4n we have 

N = (n 2 + ft! 2 )* = 5n, A r - k + ^ - 5fc - 5w, 

thus we obtain an amplitude scale which is the same as that of an oscillo- 
graph with a period constant 5n and a damping constant 5 A 1 . 

1-48. Cauchy^s method of solving a linear equation*. Let us suppose 
that we need a particular solution of the linear differential equation 

<f>(D)u = f(t), ...... (I) 

where (f> (D) = a i> + ^D"- 1 4- ... a n , 

and D denotes the operator djdt. The coefficients # s are either constants or 
functions of t. For convenience we shall write a (t) = l/a . 

If (v 1} v 2 , ... # n ) are distinct solutions of the homogeneous equation 



the coefficients C l , C 2 , . . . C n in the general solution 

t> = ^1^1 -f ^2^2+ G n V n 

may be chosen so that v satisfies the initial conditions 

v ( r ) = v' (r) = ... ^^~ 2 > (r) - 0, v^- 1 * (T) = a (T)/ (T), 

where i; (s) () denotes the ,sth derivative of v (t) and r is the initial value of 
t. We denote this solution by the symbol v (t, r) and consider the integral 



u (t) - [ v (t, r) dr. 

Jo 



Assuming that the differentiations under the integral sign can be made 
by the rule of Leibnitz, we have 



rt 
=\ 

Jo 



= 1,2, ...n- 2,n- 



the terms arising from the upper limit vanishing on account of the pro- 
perties of the function v. On the other hand 



D n u = D"v (t, r)dr-\-a (t)f(t), 

Jo 

and so <f> (D) u = / (t) + f V (D) v . dr = / (t). 

Jo 

This particular solution is characterised by the properties 

u (0) - u' (0) = ... tt<"-i> (0) - 0, w (n) (0) = a (O)/ (0). 
If, when / (0 = 1> ^ (^ T) == e/r (, r), the general value for an arbitrary 

* Except for some slight modifications this presentation follows that of F. D. Murnaghan, 
Bull. Amer. Math. Soc. vol. xxxm, p. 81 (1927). 



58 The Classical Equations 

function / (t) is seen to be / (t) if/ (t, T) and so we may write u (t) in the 
form t 

u(t)=\ t(t,T)f(t)dr. ...... (II) 

Jo 

Introducing the notation 



= J 



we have - g (t, r) = - iff (t, T), 

and the integral in (II) may be integrated by parts giving 



u (t) =/(0) f (t, 0) + f (*, r)/' (r) dr. 

Jo 

This is the mathematical statement of the Boltzmann-Hopkinson 
principle of superposition, according to which we are able to build up a 
particular solution of equation (I) from a corresponding particular solution 
(t, r) for the case in which 

/W=l t>*> 
= t< T. 

When the coefficients in the polynomial </> (D) are all constants we may 
Write a (D - r t ) (D - r,) . . . (Z> - rj, 



where r 1? r 2 , ... r n are the roots of the algebraic equation <f> (x) = 0. Taking 
first the case in which these roots are all distinct, we write 

v s = er- T > s = 1, 2, ... n. 

The equations to determine the constants C are then 



We may solve for C by multiplying these equations respectively by 
the coefficients of the successive powers of x in the expansion 

( / y* V \ ( *Y V \ If , ff \ /j - 1 , r) <> I h /jW 1 

x r 2 ) (x r 3 ) ... (x r n ) o -t- u^x -t- ... v n -iX 

Since b n _ t = 1 we find that 

/ (^ = Ci (T! - r 2 ) (r - r 3 ) ... fa - r n ) = aCtf fa), 
and the other coefficients may be determined in a similar way. 
Writing k r for the reciprocal of <' (r s ) we have 

00 

v (f -\ _ f /-\ V ^ ^^(i-r) _ /" /-.\ ./. // -\ 
v \ l '> T / y V T / ^ ^s e 7 \ T / T \^y T )> 

8-1 

and it p s denotes the reciprocal of r 8 

n 

g (t, r) = 2 p a k a [e r s (i ~ r) 1]. 

a-l 



Heaviside's Expansion 59 

i n if 



and so [< (O)]- 1 - - S p^, 



When there is a double root r l = r 2 , we write 

^ = e r i- T >, v 2 - (t - T) e f i- T) , 

v 3 , ... v n being the same as before. The equations to determine the constants 
Carenow <7 4 + . <7, + C7, + ... 0. = 0, 

r^ + 1 . <7 2 + r 3 . (7 3 + ... r n . C n = 0, 
/y-i/7, + (-!) r.-^C, + rj-'C 1 , + ... r,,--^, = af (t). 
Writing (r - A) F (r) = (r - A) (r - r 3 ) ... (r - r n ) 

= c + c t r + ... c^^"- 1 

and multiplying the equations by c , c lt ... c n _ t respectively we find that, 
since C n _ r = 1, 

G 1 (r, -\)F (r,) + C z [F (r,) + (r, - A) F' (r,)] = af (r). 

The quantity A is at our disposal. Let us first write A = r 1} we then 

have /-i n / \ f i , 

C t F (rj = af(r). 

Simplifying the preceding equation with the aid of this relation we 

btain C,F (r,) + C 2 F' (r,) = 0. 

Writing G (r) = ajF (r), 

we have C, = / (T) G' (r,), C t =f (T) O (r,). 

These are just the constants obtained by writing 



<f>(r) r-fj (r-rtf r - r 3 ">-r n ' 

and a similar rule holds in the case of a multiple root of any order or any 
number of multiple roots. Thus in the case of a triple root, 



1-49. Heaviside's expansion. The system of differential equations (II) 
of 1-47 may be written in the form 



2 a f .C.=X(0, 



where the a's are analogous to the operators ~rs of 1-45. 



60 The Classical Equations 

Denoting the determinant | a r<s \ by <f> (D) and using A rs to denote the 
co-factor of the constituent a rs in this determinant, we have 

< (D) Q, = 2 A rs E r (t). 

r=l 

To obtain an expansion for Q s we first solve the equation 

j (D) y s (t) = 1 
with the supplementary conditions 

y,(0) = y/(0) = ...y.<-(0)=o, 
then x rs = A sr y 3 (t) 

is a particular solution of the equation </> (D) x r A sr . 1 for which 

a: r (0) = * r '(0) = 0; 

and by the expansion theorem of 1-48, 
- x rs (t, 0) = A sr y s (t, 0) 

= s A_.(!-Ler.. + ^- > 
jT,(r a )^(r a ) e + .(0) ' 

which is Heaviside's expansion formula. The corresponding formula for 

<UOi- 

& (0 = S # r (0) * ir (*, 0) + tf/ (r) * sr (t, r) dr , 

r-l L JO J 

and this particular solution satisfies the conditions 

Q s (0) = Q.' (0) = o. 

1-51. The simple wave-equation. There are a few partial differential 
equations which occur so frequently in physical problems that they may 
be called classical. The first of these is the simple wave-equation 



which occurs in the theory of a vibrating string and also in the theory of 
the propagation of plane waves which travel without change of form. 
These waves may be waves of sound, elastic waves of various kinds, waves 
of light, electromagnetic waves and waves on the surface of water. In 
each case the constant c represents the velocity of propagation of a phase 
of a disturbance. The meaning of phase may be made clear by considering 
the particular solution 

V = sin (x ct) 

which shows that F has a constant value whenever the angle x ct has 
a constant value. This angle may be called the phase angle, it is constant 
for a moving point whose x-co-ordinate is given by an equation of type 
x = ct -I- a, where a is a constant. This point moves in one direction with 
uniform velocity c. There is also a second particular solution 

V = sin (x -\- ct) 



Wave Propagation 61 

for which the phase angle x -f ct is constant for a point which moves with 
velocity c in the direction for which x decreases. These solutions may be 
generalised by multiplying the argument x ct by a frequency factor 
27TV/C, where v is a constant called the frequency, by adding a constant y 
to the new phase angle and by multiplying the sine by a factor A to 
represent the amplitude of a travelling disturbance. In this way the 
particular solution is made more useful from a physical standpoint be- 
cause it involves more quantities which may be physically measurable. In 
some cases these quantities may be more or less determined by the supple- 
mentary conditions which go with the equation when it is derived from 
physical principles or hypotheses. 

Usually this particular equation is derived by the elimination of the 
quantity U from two equations 

dv du zu _ dv ... 

~di~ a dx' ~dt~ p fa ...... IA) 

involving the quantities U and F, the coefficients a and ft being constants. 
The constant c is now given by the equation 

c 2 - aft. 

It should be noticed that if F is eliminated instead of 17, the equation 
obtained for U is 



and is of the same type as that obtained for V. This seems to be a general 
rule when the original equations are linear homogeneous equations of the 
first order with constant coefficients, however many equations there may 
be. The rule breaks down, however, when the coefficients are functions of 
the independent variables. If, for instance, a and ft are functions of x the 
resulting equations are respectively 

327 a 



cw . a / du\ 

dt 2 dx \ dx / ' 

These equations may be called associated equations. Partial differential 
equations of this type occur in many physical problems. If, for instance, 
y denotes the horizontal deflection of a hanging chain which is performing 
small oscillations in a transverse direction, the equation of vibration is 

8 2 y __ a / dy\ 



where g is the acceleration of gravity and x is the vertical distance above 
the free end. Equations of the above type occur also in the theory of the 
propagation of shearing waves in a medium stratified in horizontal plane 
layers, the physical properties of the medium varying with the depth. 



62 The Classical Equations 

1-52. The differential equation (I) was solved by d'Alembert who 
showed that the solution can be expressed in the form 

V=f(x-ct) + g(x + ct), 

where / (z) and g (z) are arbitrary functions of z with second derivatives 
/" ( z )> $" ( z ) that are continuous for some range of the real variable z. 
A solution of type f (x ct) will be called a " primary solution," a term 
which will be extended in 1-92 to certain other partial differential 
equations. 

To illustrate the way in which primary solutions can be used to solve 
a physical problem we consider the transverse vibrations of a fine string 
or the shearing vibration of a building*. 

The co-ordinate x is supposed to be in the direction of the undisturbed 
string and in the vertical direction for the building, the co-ordinate y is 
taken to represent the transverse displacement. If A denotes the area of 
the cross-section, which is a horizontal section in the case of the building, 
and p the density of the material, the momentum of the slice Adx is M dx, 

where M pA ^ . The slice is acted upon by two shearing forces acting 

in a transverse direction and by other forces acting in a "vertical" 
direction, i.e. in the diiection of the undisturbed string. Denoting the 

shearing force on the section x by S, that on the section x -f dx is S -f ~ dx. 

dS dx 

The difference is x dx, and so the equation of motion is 
ox 

m __ ss 

St ~dx' 

We now adopt the hypothesis that when the displacement y is very 
small ~ 



where /JL is a constant which represents the rigidity of the material in the 
case of the building and the tension in the case of the string. According 
to this hypothesis if p and A are also constants 



where c 2 = p/p. The expressions for M and S also give the equation 

as BM 
p-ft-P-to' 

and so we have two equations of the first order connecting the quantities 
M and 8', these equations imply that M and S satisfy the same partial 
differential equation as y. 

i 

* The shearing vibrations of a building have been discussed by K. Suyehiro, Journal of the 
Institute of Japanese Architects, July (1926). 



Transmission of Vibrations 63 

In the case of the building one of the boundary conditions is that there 
is no shearing force at the top of the building, therefore 8 = when x = h. 
Assuming that 

y=f(x-ct) + g(x + ct) 

di/ 
the condition is ^ = when x = h. and so 

ox 

= /' (A - ct) + g' (h + ct). 
This condition may be satisfied by writing 

y = <f> (ct + x h) ~h (f> (ct x + 7^), 

where </> (2) is an arbitrary function. 

A motion of the ground (x = 0) which will give rise to a motion of this 
kind is obtained by putting x = in the above equation. 

Denoting the motion of the ground by y = F (t) we have the equation 

F (t) = <f> (ct - h) -\- </> (ct 4- h) ...... (A) 

for the determination of the function </> (z). 

In the case of the string the end x = / may be stationary. We therefore 
put y = for x = I and obtain the equation 

0=f(l-ct) + g(l + ct) 
which is satisfied by 



where $ is another arbitrary function. If the motion of the end x = is 
prescribed and is y = (t) we have the equation 

-0 (t) = </r (Ct - 1) - $ (Ct + I) ...... (B) 

for the determination of the function $ (z). 

If, on the other hand, the initial displacement and velocity are pre- 
scribed, say ~ 

y=e(x), gf = *() 

when t = 0, we*have the equations 

0(x)=f (x) + g (x), x (x) = c [<?' (x) - /' (x)] 
which give 2cf (x) = c9' (x) - x( x )> 

2cg' (x) = cff (x) + x (x), 
and the solution takes the form 

1 rx+ct 

y=\\0(x~ct) + e(x + ct)-\+ f \ x (T)dr. 

L(j J x - ct 

If in the preceding case both ends of the string are fixed, the equation 
(B) implies that ^ (x) is a periodic function of period 21, the corresponding 
time interval being 2l/c. Submultiples of these periods are, of course, 
admissible, and the inference is that a string with its ends fixed can perform 
oscillations in which any state of the system is repeated after every time 



64 The Classical Equations 

interval of length 2lm/nc, where m and n are integers, n being a constant 
for this type of oscillation. 

In the case of the building the ground can remain fixed in cases when 
(f> (z) is a periodic function of period 4=h such that 

<f) (z -f 2h) = <f) (z). 

It should be noticed that the conditions of periodicity may be satisfied 
by writing 

(z) = sin - j- , (f> (z) = sin [(n + |) TTZ/&], 

where in and n are integers. Thus in the case of the string with fixed ends 
there are possible vibrations of type 



mirx irnrct 
- cos 



y-~, 

and in the case of the building with a free top and fixed base there are 
possible vibrations of type 

. . r/ iv 7 ^"! r/ ivfrttfi 

y = b m sin \(n + J) , | cos \(n -f J) -,- . 

L ^ J L ^ J 

These motions may be generalised by writing for the case of the string 

. mnx mnct 
y - S a m sin - cos y , 

m-l * * 

where the coefficients a w are arbitrary constants. For complete generality 
we must make s infinite, but for the present we shall treat it as a finite 
constant. The total kinetic energy of the string is 



, [i . mnx . HTTX -, 

since we have sm 7 sm . ax = n + m 

Jo * * 

= Z/2 ft = m. 

Since the kinetic energy is the sum of the kinetic energies of the motions 
corresponding to the individual terms of the series, these terms are supposed 
to represent independent natural vibrations of the string. These are generally 
called the normal vibrations. 

The solution for the vibrating building can also be generalised so as to 
give 3 

y= S b n sin \(n -f J) \- cos \ (n -f i) -,- L 
n-i L ^J L h \ 

and the kinetic energy is in this case 

f, (271-f 1) 2 7T 2 C 2 , 2 . 

> ^ ' *x 2 nt v^ 



Vibration of a String 65 

for now we have corresponding relations 

/ sin \(n 4- J) ^- sin (m -f J) ^- ute = mn 

= A/2 m n. 
Such relations are called orthogonal relations. 

1-53. In the case of the equation of the transverse vibrations of a 
string there is a type of solution which can be regarded as fundamental. 
Let us suppose that the point x = a is compelled to move with a simple 

harmonic motion* n , t . 

y = p cos (pt 4- ), 

where a, /3 and p are arbitrary real constants. If the ends x = 0, x = I 
remain fixed, it is easily seen that the differential equation 



W dx* ' 

and the conditions y = at the ends may be satisfied by writing y y for 
< x < a and y = y 2 for a < x < /, where 

y l = p cosec (Xa) sin (\x) cos (p -fa), 

2/2 = /? cosec A (/ a) sin A (/ #) cos (pt + a), 

and Ac = p. The case of a periodic force F = F cos (^tf -f a) concentrated 
on an infinitely short length of the string may be deduced by writing down 
the condition that the forces on the element must balance, the inertia 
being negligible. This condition is 

F = Py/ - PyJ for x = a, 
where P is the pull of the string. Substituting the values of y^ and y% we 

get cF = pfiP [cot Aa + cot A (I - a)] . 

F 
Therefore /? = p^ cosec XI sin Aa sin A (I a). 

The solution can now be written in the form 

F 

y= j>0 (*>)> 

, , . sin A# sin A (Z a) 

where <7 (x, a) = - c ^ ^ -- - < x < a 

A sin AL 

_ sin A (I x) sin Aa 7 ' 

= ----- x - . ^j ----- a ^ x ^ /. 
A sin XI 

This function gr (a;, a) is a solution of the differei^tial equation 

S+*V-0 ...... (A) 

* Rayleigh, Theory of Sound, vol. I, p. 195 



66 The Classical Equations 

and satisfies the boundary conditions g = when x and when x I. 
It is continuous throughout the interval < x < I, but its first derivative 
is discontinuous at the point x = a and indeed in such a manner that 

lim 

8 ->Q 

The function g (x, a) is called a Green's function for the differential 

d 2 u 
expression , ^ 4- A 2 w, it possesses the remarkable property of symmetry 

expressed by the relation 

g (x, a) = g (a, x). 

This is a particular case of the general reciprocal theorem proved by 
Maxwell and the late Lord Rayleigh. 

It should be noticed that the Green's function does not exist when A 
has a value for which sin XI = 0? that is, a value for which the equation (A) 
possesses a solution g = sin Xx which satisfies the boundary conditions and 
is continuous (D, 1) throughout the range (0 < x < /). 

A fundamental property of the Green's function g (x, a) is obtained by 
solving the differential equation 



by the method of integrating factors. Assuming that y is continuous (Z>, 2) 
in the interval (0, 1) and that the function/ (x) is continuous in this interval, 
the result is that 



U = U - U 

a-0 



~ 

l Jo 



where u (0) and u (I) are assigned values of u at the ends. If these values 
are both zero y is expressed simply as a definite integral involving the 
Green's function and/ (a). 

1-54. The torsional oscillations of a circular rod are very similar in 
character to the shearing oscillations of a building. Let us consider a 
straight rod of uniform cross-section, the centroids of the sections by 
planes x = constant, perpendicular to the length of the rod, being on a 
straight line which we take as axis of x. Let us assume that the section at 
distance x from the origin is twisted through an angle 6 relative to the 
section at the origin. It is on account of the variation of 8 with x that 
an element of the rod must be regarded as strained. The twist per unit 
length at the place x is defined to be 

B8 

T = ^ ; 
it vanishes when 8 is constant throughout the element bounded by the 



Torsional Oscillations 67 

planes x and x -f dx, i.e. when this element is simply in a displaced position 
just as if it had been rotated like a rigid body. 

The torque which is transmitted from element to element across the 
plane x is assumed to be Kfir, where /A is an elastic constant for the material 
(the modulus of rigidity) and K is a quantity which depends upon the size 
and shape of the cross-section and has the same dimensions as 7, the 
moment of inertia of the area about the axis of x. 

Let p denote the density of the material, then the moment of inertia 
abou^ the axis of x of the element previously considered is pldx and the 
angular momentum is plOdx. 

Equating the rate of change of angular momentum to the difference 
between the torques transmitted across the plane faces of the element, we 
obtain the equation of motion 

. d 2 6 cW 
pi = fih 
cl 2 ^ fix* 

which holds in the case when the rod is entirely free or is acted upon by 
forces and couples at its ends. In this case the differential equation must 
be combined with suitable end conditions. 

A simple case of some interest is that in which the end x is tightly 
clamped, whilst the motion of the other end x a is prescribed. 

1-55. The same differential equation occurs alstfin the theory of the 
longitudinal vibrations of a bar or of a mass of gas. 

Consider first the case of a bar or prism whose generators are parallel 
to the axis of x. Let x -f denote the position at time t of that cross- 
section whose undisturbed position is x, then denotes the displacement 
of this cross-section. An element of length, 8x, is then altered to 8 (x -f- f ), 
of (1 -f ') $x, where the prime denotes differentiation with respect to x. 
Equating this to (1 + e) $x we shall call e the strain. The strain is thus the 
ratio of the change in length to the original length of the element and is 
given by the formula ^ 

e = ' 

dx 

According to Hooke's law stress is proportional to strain for small 
displacements and strains. The total force acting across the sectional area 
A in a longitudinal direction is therefore F = EeA, where E is Young's 
modulus of elasticity for the material of which the rod is composed. The 
stress across the area is simply Ee. 

The momentum of the portion included between the two sections with 

co-ordinates x and x -f 8x is M 8x, where M = pA ~ and p is the density of 
the material. The equation of motion is then 

<W W 
dt ~ dx 9 

5-2 



68 The Classical Equations 

A S * 

A ' 



When the material is homogeneous and the rod is of uniform section the 
equation is ^ 

a^^cte 5 ' 

where c 2 = //o. Since the modulus E for most materials is about two or 
three times the modulus of rigidity /z, longitudinal waves travel much more 
rapidly than shearing waves and the frequency of the fundamental mode 
of vibration is higher for longitudinal oscillations than it is for shearing 
oscillations. In the case of a thin rod shearing oscillations would not occur 
alone but would be combined with bending, and the motion is different. 

The fundamental frequency for the lateral oscillations is, however, much 
lower than that for the longitudinal oscillations. Let us next consider the 
propagation of plane waves of sound in a direction parallel to the axis of x. 

Let VQ = A 8x be the initial volume of a disc-shaped mass of the gas 
through which the sound travels, v = A8 (x -f ) the volume of the same 
mass at time t. We then have 

v = v Q (l -f e), 

where e is now the dilatation. If p Q is the original density and p the density 
of the mass at time t, we may write 

P = PQ (1 + s), 

where s is the condensation, it is the ratio of the increment of density to 
the original density. Since pv = p Q v Q we have 

(1 + s) (1 + e) = 1, 
and if s and e are both small we may write 



~ . 
ox 

To obtain the equation of motion we assume that the pressure varies 
with the density according to some definite law such as the adiabatic law 



where p is the pressure corresponding to the density y and is a constant 
which is different for different gases. 

This law holds when there is no sensible transfer of heat between 
adjacent portions of the gas. Such a state of affairs corresponds closely 
to the facts, since in the case of vibration of audible frequency the con- 
densations and rarefactions of our disc-shaped mass of gas follow one 
another with a frequency of 500 or more per second. 

For small values of s we may write 

P = Po( l + 7 s )- 



Waves of Sound 69 

The equation of motion is now 

m _SF 

St ~ Sx ' 
where M = Po A ^, F = - Ap. 

Substituting the values of p and s we obtain the equation 

af_ af 

~ 



in which c = y 

Po 

For sound waves in a tube closed at both ends the boundary conditions 
are =. when x = and when # = I. The solution is just the same as the 
solution of the problem of transverse vibration of a string with fixed ends. 

For sound waves in a pipe open at both ends and for the longitudinal 
vibrations of a bar free at both ends we have the boundary conditions 



when x = and when x = 1 9 which express that there is no stress at the 
ends. The normal modes of vibration are now of type 

.. ~ nnrx (rmrct\ 

g=C m COS -j- COS ^-y- J , 

where C m is an arbitrary constant and m is an integer. This solution is of 
type = <l>(x + ct) + <f>(ct- x), 

and may be interpreted to mean that the progressive waves represented 
by ^ = <f> (ct x) are reflected at the end x = with the result that there 
is a superposed wave represented by > = < (ct -f x). 

There is a different type of reflection at a closed end of a tube (or fixed 
end of a rod), as may be seen from the solution 

= ^ (ct - x) - <f> (ct + x), 

which makes f = when x = 0. 

Reflection at a boundary between two different fluid media or between 
two parts of a bar composed of different materials may be treated by 
introducing the boundary condition that the stress and the velocity must 
be continuous at the boundary. 

If progressive waves represented by = # </> (t x/c) approach the 
boundary x = from the negative side and give rise to a reflected wave 
x = a^ (t 4- x/c) and a transmitted wave 2 = ^2^ (^ "~ #/ c/ )> the boundary 
conditions- are fit fit fit 

' 



" ** 



** 



70 The Classical Equations 

where K = yp and K = y'p , the constants y and y' referring respectively 
to the media on the negative and positive sides of the origin. The equi- 
librium pressure p Q is the same for both media. 

9& 9fi 9 & 

Now -* = , -g^ =-<i, * = c'*, 

hence, when x = 0, 

c$ - o^' (0, - cs! = <*!</>' (t), c'3ij= a^ (t), 
and c (s s^) = c's 2 , K (S Q -\- Sj) i<'s. 



mi . 

Iherefore s l s $ = - 5 

1 jc'c-f KC' 2 //C-f ice' ' 



KC 



t , , / 

KC+ KC' /c'C+ ACC 7 

1-56. The simple wave-equation occurs also in an approximate theory 
of long waves travelling along a straight canal, with horizontal bed and 
parallel vertical sides, the axis of x being parallel to the vertical sides and 
in the bed (see Lamb's Hydrodynamics, Oh. vm). 

Let 6 be the breadth of the canal and h the depth of the fluid in an 
initial state at time t when the fluid is at rest and its surface horizontal. 
We shall denote the density of the fluid by p and the pressure at a point 
(x, y, z) by p. The motion is investigated on the assumption that p is 
approximately the same as the hydrostatic pressure due to the depth 
below the free surface. This means that we write 

P = Po + 9P (h + ri- y), ...... (I) 

where ^p Q is the external pressure, which is supposed to be uniform, 77 is 
the elevation of the free surface above its undisturbed position and g is 
the acceleration of gravity. One consequence of this assumption is that 
there is no vertical acceleration, in other words, the vertical acceleration is 
neglected in making this approximation. 

If, in fact, we consider a small element of fluid bounded by horizontal 
and vertical planes parallel to the planes of reference, the axis of y being 
vertically upwards, the equations of motion are 

pa . Sx8ySz = ~ 8x . 8y$z, 

pft . 8x8ySz = ~ - Sy . 8z8x pg8x8y8z, 

d%) 
py . 8x8y8z = 8z . 8x8y, 

where a, j8, y are the component accelerations. With the above assumption 

we have 8 = y = 0, and so ^ 

op 



Waves in a Canal 



71 



The assumption of no vertical acceleration is not equivalent to the 
assumption (I), because an arbitrary function of x, z and t could be added 
to the right-hand side of (I) and the equations of motion would still give 
no vertical acceleration. 

Equation (I) gives pa = gp x- . 

This expression for 2 is independent of y, consequently, since g is 
assumed to be constant, the acceleration a is the same for all particles in 
a vertical plane perpendicular to the axis of x. The horizontal velocity u 
depends on x and t only. 

Now let be the total displacement from their initial position of the 
particles which at time t occupy the vertical plane x. Each particle is 
supposed to have moved horizontally through a distance , but actually 
some of the particles will have moved slightly upwards or downwards as well. 

the fluid which occupies the region QQ'X'X is 
supposed to have initially occupied the region 
PP'A'A. 

Equating the amount of fluid in the region 
QQ'N'N to the difference of the amounts in the 
regions PNXA, P'N'X'A' we obtain the equa- 
tion of continuity 



N 



'N' 



A A X X' 

Fig. 9. 



or 



i 

= - h - . 

dx 



.(II) 



A second equation is obtained by writing a = >, . This is approximately 

ot 

true in the case of infinitely small motions, the exact equation being 



a 



du Bu 



u 



dt^"dx 

?=!*&, 

du 



Writing 

we have |^ 2 = ~~ = a = - g g. (Ill) 

The equations (II) and (III) now give the wave-equations 

where c 2 = gh. 

When, in addition to gravity, the fluid is acted upon by small dis- 
turbing forces with components (X, Y) per unit mass of the fluid, the 



72 The Classical Equations 

assumption that the pressure is approximately equal to the hydrostatic 
pressure leads to the equation 

- 

(g-Y)dy. 



v 



TU- P i vx*7 

Thisgives -/>&- Y ^x 

and the equation of horizontal motion 

du dp 

'-fc-^-fc 

indicates that in general u depends on y as well as on x and t. 

With, however, the simplifying assumptions that Y is small compared 

dY 
with g and that h ~ is small in comparison with X the equation takes the 

form 3 _ 



and, if X depends only on x and t, this equation indicates that u is inde- 
pendent of y. We may then proceed as before and obtain the equations 



EXAMPLES 

1. An elastic bar of length I has masses m , m 1 at the ends x 0, x I respectively. 
Prove that the terminal conditions are 

Jj] A - _ vn wVi (*T\ y ' . 

^^ o- ~ m Q 1*9. wiit-ii & v, 



Prove that the possible frequencies of vibration are given by the equation 

(1 - KHiP) tan 9 + U + /*i) ^ = 0, 

where c 2 m^ = lAE^ t c 2 m l = lAE^i lt = nl, 

and nc/27T is the number of vibrations per second. 

2. If a prescribed vibration = C cos w is maintained at the end x = of a straight 
pipe which is closed at the end x = I the vibration at the place x is given by 

~ nl . n (I x) 

f = C cosec - sin cos nt. 

c c 

Obtain the corresponding solution for the case in which the end x = I is open. 

3. Discuss the longitudinal oscillations of a weighted bar whose upper end is fixed. 



Systems of Equations 73 

4. If ^ 



and A is an arbitrary constant, the function 

y A [sin 2sna sin 

satisfies the differential equation _ _ 

d*y _ d*y 

2 ~ z ' 



and the end conditions y when x = and when x = a + vt. Prove also that when v -> 0, 



, . SnX Snct 

it -> 2 A sm cos . 
a a 



[T. H. Havelock, Phil. Mag. vol. XLVH, p. 754 (1924).] 
5. Prove that if y when x = and x = vt, 
y =/(*). y = 9 (*)> 
a solution of ~ ^ = c 2 ^~ is given by 



~ ^ 



where a log -- = 2n, 

exp (r) = ~ - -- ^K + ) = i/() -f I f X g(x)dx, 

fo z C z o ^ c .' 



B n - - - f * f K + *) cos (naa>) -^- - , 

7T J - V t MO + 3? 

and it is supposed that . . . . , % 

FF ^ /(-*) = -/(*), g(-*)--^(). 

[E. L. Nicolai, PM. Jfogr. vol. XLIX, p. 171 (1925).] 



1-61. Conjugate functions and systems of partial differential equations. 
If in equations ((A) 1-51) we write a = 1, /? = - 1 and use the variable 
y in place of t we obtain the equations 

W^dV dU^^dV 

dx dy y dy ~ ~dx 

satisfied by two conjugate functions U and V. In this case both functions 
satisfy the two-dimensional form of Laplace's equation 



_ 

dx* 3^2 ~ - 

This equation is important in hydrodynamics and in electricity and 
magnetism. 

The equations (A) may be generalised in another way by writing 

.97 dU 3V 




74 The Classical Equations 

where a, /?, y, 8, 6, </>, A, /x, a, r are arbitrary constants. In particular, 

the equations w ^V dU 

dt ==SK 'dx' ~dx^ V 



i A+ ^ f dU 

lead to the equation -~-- = K ~ 2 , 

which is the equation for the conduction of heat in one direction when U 
is interpreted as the temperature and K as the diffusivity. The same 
equation occurs in the theory of diffusion. It should be noticed that the 
quantity V satisfies the same equation. 

Again, if we write -~ = L -x- -f RU, 

du ^c dv + sv 
dx dt ^ ' 

and interpret V as electric potential, U as electric current, we obtain the 
differential equation 



which governs the propagation of an electric current in a cable*. The 
coefficients have the following meanings : 

R L C - S 

resistance inductance capacity leakance 

all per unit of length of the cable. The quantity U satisfies the same 

differential equation as V. This differential equation may be reduced to a 

canonical form by introducing the new dependent variables u, v, defined 

by the equations TT mlT T7 p . /r 

J ^ u = Ue Rt i L , v = Ve m < L . 

These variables satisfy the equations 

dv ,- du 



and the canonical equations of propagation are Heaviside's equations 



These equations are of the simple type (I) if 

SL = CR. 
In this case a wave can be propagated along the cable without distortion. 

* Cf . J. A. Fleming, The Propagation of Electric Currents in Telephone and Telegraph Circuits, ch. v. 



The Telegraphic Equation 75 

When dealing with the general equations (A) it is advantageous to use 
algebraic symbols for the differential operators and to write 

3 n d -D 
si n " te~ Dx ' 

the differential equations may then be written symbolically in the form 
(0D t - yD x - /*) F = (aD. + A) U, (<f>D t -W x -a)U = (fiD x + r) V. 
The first equation may be satisfied by writing 

U=(OD t -yD.-rtW, V=(aD x +\)W, ...... (B) 

where W is a new dependent variable. Substituting in the second equation 
we obtain the following equation for W, 

[(0D t - yD x - p) (^D t - 8D X - a) ~ (aD x + A) ($D X +r)]W^ 0, 

which, when written in full, has the form 
9217 9217 921^ 

/v / f rr / n& , i \ V r " . / & /*>\ U " i f\ 



- (ar -f j3A + ycr + 8/i) ~ + (AIO- - XT) W = 0. 

When this equation has been solved the variables U and V may be 
determined with the aid of equations (B). It is easily seen that U and V 
satisfy the same equation as W. 

The equation for W is said to be hyperbolic, parabolic or elliptic 
according as the roots of the quadratic equation 

6</>X* - (68 + fa) X + yS - aft - 

are real and distinct, equal or imaginary. In this classification the co- 
efficients a, /?, y, S, B y fi, A, /z, a, r are supposed to be all real, the simple 
wave-equation is then of hyperbolic type, the equation of the conduction 
of heat of parabolic type and Laplace's equation of elliptic type. The 
telegraphic equation is generally of hyperbolic type, but if either C = or 
L = it is of parabolic type and the canonical equation is of the same form 
as the equation of the conduction of heat. 

The foregoing analysis requires modification if the coefficients a, j8, y, 
8, 0, <f>, A, /A, a, r are functions of x and t, because then the operators aD x -f- A 
and 9D t yD x \L are not commutative in general, and so the first 
equation cannot usually be satisfied by means of the substitution (B). If, 
however, the conditions ^^ 



da 



76 The Classical Equations 

are satisfied the operators are commutative (permutable) and a differential 
equation may be obtained for W. In this case the variables U and F do 
riot necessarily satisfy the same partial differential equation. This is easily 
seen by considering the simple case when the first equation is U = 0dV/dt 
and /? and r are independent of t. 

Differential operators which are not permutable play an interesting 
part in the new mechanics. 

1-62. For some purposes it is useful to consider the partial difference 
equations which are analogous to partial differential equations in which 
we are interested. The notation which is now being used in Germany is 
the following * : 

u (x + h, y) - u (x, y) = hu x , u (x, y -f h) - u (x, y) = hu v , 
u (x, y} u (x h, y) = hu^, u (x, y) u (x, y h) = hu^, 

u (x + h,y) 2u (x, y) -f u (x h, y) = Ji 2 u x ^ = h^u^x - 
The equations u x Vy> u y v% 

are analogous to those satisfied by conjugate functions since they imply 

U x7 + V v y = 0, V x * + Vyy = 0. 

The equations % v v , u^ v x 

give the equations u x2 = u$ , v x ^ = v g 

analogous to the equation of the conduction of heat. 

1-63. The simultaneous equations from which the final partial 
differential equation is derived need not be always of the first order. In 
the theory of the transverse vibrations of a thin rod the primary equations 



where 77 is the lateral displacement,^ the bending moment, A the sectional 
area, x the radius of gyration of the area of the cross-section about an axis 
through its centre of gravity, p the density and E the Young's modulus 
of the material. The resulting equation 



is of the fourth order. The equation is usually simplified by the omission 
of the second term. This process of approximation needs to be carefully 
justified because it will be noticed that the term omitted involves a 
derivative of the fourth order, that is a derivative of the highest order. 
Now there is a danger in omitting terms involving derivatives of the highest 

* See an article by R. Courant, K. Friedrichs and H. Lewy, Math. Ann. Bd. c, S. 32 (1928). 
f Cf. H. Lamb, Dynamical Theory of Sound, p. 121. 



Vibration of a Rod 77 

order because their coefficients are small. This may be illustrated in a very 
simple way by considering the equation 



where v is small. The solution is of type 

7] = A -f JBe*>, 

where A and B are constants. When the term on the right of (IT) is omitted 
the solution is simply 77 A. When x and v are both small and positive 
the term Be x l v , which is omitted in the foregoing method of approximation, 
may be really the dominant term. In this example all the terms involving 
derivatives of the highest order have been omitted, and as a general rule 
this is more dangerous than the omission of only some of the terms as in 
the case of the vibrating rod. The omission of the second term from the 
rod equation seems to be quite justifiable when the rod is very thin. When 
the rod is thick Timoshenko's theory* shows that there is a term giving 
the correction for shear which is at least as important as the second term 
of the usual equation (I). 

This point relating to the danger of omitting terms involving derivatives 
of the highest order comes up again in hydrodynamics when the question 
.of the omission of some or all of the viscous terms comes under considera- 
tion. The omission of all the viscous terms lowers the order of the equations 
and requires a modification of the boundary conditions. This does riot lead 
to very good results. On the other hand, in PrandtFs theory of the 
boundary layer some of the viscous terms are retained, the boundary 
condition of no slipping at the surface of a solid body is also retained and 
the results are found to be fairly satisfactory. 

EXAMPLE 

r A u j. ^u A - & v Bu , du 

Prove that the equations - = a - - -f b ~- , 

ox ox oy 

dv du , du 
2 = c 5- -f d 5- 
oy ox oy 

give an equation of the second order which is elliptic, parabolic or hyperbolic according as 
(a d) 2 + 46c is less than, equal to or greater than zero. 

[E. Picard, CompL Rend. t. cxn, p. 685 (1891).] 

1-71. Potentials and stream-functions. The classical equations are of 
great mathematical interest and have played an important part in the 

* Phil. Mag. (6), vol. XLI, p. 744 (1921). The equation used by Timoshenko is of type 

2 3S? d*T) 2 
EK 3~4 + P aTa - P" 

* r * 



where jz is the mocjultis of rigidity and a is a constant which depends upon the shape of the cross - 
section. For the equation of resisted vibrations see Note II, Appendix. 



78 The Classical Equations 

development of mathematical analysis by suggesting fruitful lines of 
investigation. It can be truly said that the modern theory of functions 
owes its origin largely to a study of these equations. The theory of functions 
of a complex variable is associated, for instance, with the theory of con- 
jugate functions and the solutions of Laplace's equation. 

If, for instance, we write 

+ ty=/(a; + ty) =/(), 

where/ (z) is an analytic function* and (f> and are real when x and y are 
real, we have, for points in the domain for which/ (z) is analytic, 



= 
dy dy J 

where f (z) denotes the derivative of /(z). 
These equations give 



Equating the real and imaginary parts of the two sides of this equation, 
we see that ~ . - , 



d<f> dJj 

V= - a r = ^, say. 

oy ox J 

These relations between the derivatives of two conjugate functions <f> 
and ifj are called Cauchy's relations because they play a fundamental part 
in Cauchy's theory of functions of a complex variable. The relations can 
also be given many very interesting physical interpretations. 

The simplest from a physical standpoint is, perhaps, that in which u 
and v are regarded as the component velocities in the plane of x, y of a 
particle of a fluid in two-dimensional motion, the particle in question being 
the particular one which happens to be at the point (x, y) at time t. If u 
and v are independent of t the motion is said to be "steady" and a curve 
along which it is constant may be regarded as a "stream-line" or "line of 
flow" of the particles of fluid. The condition that a particle of the fluid 
should move along such a line is, in fact, expressed by the differential 
equations 7 , 

d ^ = d -y =d t ...... (B) 

u v v ' 

which give vdx udy = 0, 

that is dift = or if/ = constant. 

* The reader is supposed to possess some knowledge of the properties of analytic functions. 



Conjugate Functions 79 

Another way of looking at the matter is to calculate the "flux" across 
any line AP from right to left. This is expressed by the integral 



where ds denotes an element of length of AP and the suffix is used to 
indicate the point at which \fj is calculated. It is clear from this equation 
that there is no flow across a line AP along which if/ is constant. 

The conjugate function < is called the " velocity potential" and was 
first introduced by Euler. The curves on which (f> is constant are called 
" equipotential curves." The function </f is called the stream-function or 
current function, it was used in a general manner by Earnshaw. 

It must be understood that the fluid motion which is represented by 
such simple formulae is of an ideal character and is only a very rough 
approximation to a real motion of a fluid. A study of this type of fluid 
motion serves, however, as a good introduction to the difficult mathe- 
matical analysis connected with the studies of actual fluid motions. It will 
be worth while, then, to make a few remarks on the peculiarities of this ideal 
type of fluid motion*. 

In the first place, it should be noticed that the expression udx 4- vdy 
is an exact differential dcf>, and so the integral 



I (udx + vdy) 



represents the difference between the values of <f> at the ends of the path of 
integration. If the function (f> is one -valued the integral round a closed 
curve is zero, but if <f> is many-valued the integral may not vanish. The 
value of the integral in such a case is called the circulation round the 
closed curve. It is different from zero in the case when 

< -f ty = i log z = i (log r + id) 
and the curve is a circle whose centre is at the origin. In this case 

< = _ 0, = log r, 

and it is easily seen that the circulation F defined by the integral 

f r r 2zr 

r = udx + vdy = ld<f> = - dO 

J J Jo 

is equal to 2n. The fluid motion for which 

< -f ty = A log z, 

where A is a constant, is said to be that due to a vortex of strength F when 

ir 

A is an imaginary quantity - . If, on the other hand, A is real, the motion 

J-tTT 

is said to be due to a source if A is positive and due to a sink if A is 
negative. The flow in the last two cases is radial. 



80 The," Classical Equations 

Since the stream -function in the last two cases is Ad and is not one- 
valued, the flux across a circle whose centre is at is 2-n-A. 

The flow due to a vortex, source or sink at a point other than the origin 
may be represented in the same way by simply interpreting r and 6 as 
polar co-ordinates relative to the point in question. 

Since the equations expressing u and v in terms of </> and $ are linear, 
the component velocities for the flow due to any number of vortices, 
sources and sinks may be derived from the complex potential 

<f> + it- 27T s ( - *&) log IX- x * + i(y- y*)] > 

where the constants a s , j8 s specify the strengths of the source and vortex 
associated with the point (x s , y s ). The word source is used here in a general 
sense to include both source and sink. 

One further remark may be made regarding the motion if we are 
interested in the career of a particular particle of fluid. If x , y are the 
initial co-ordinates of this particle at time t these quantities at time t will 
be functions of x, y and t 

% - / (x, y, 2/0 = g (x> y, 0> 

but functions of such a nature that the equations (B) are satisfied when 
x Q and y are regarded as constant. We have then 

!/+!/+ jjf-o, *+, <*+*=, o ...... (c.D) 

dx oy dt ox oy ot ' 

and any quantity h which can be expressed in the form h = F (X Q , y ) will 
be a solution of the equation 

dh dh dh 

a. + M 3 + V 5 - = 0, 

dt dx dy 

and will be constant throughout the motion. We shall write this equation 
in the form dhjdt = and shall call dh/dt the complete time derivative of 
h. When the motion is steady we evidently have difj/dt = 0. 

The equations (C) and (D) may be solved for u and v if -- = 1 and 

o (Xj yj 



give expressions *~ 

which satisfy the equation 



= 

ox dy 



on account of - > = 1. 

a (x, y) a (x, y) 

This last equation expresses that the area occupied by a group of 
particles remains constant during the motion. To obtain a solution of this 
equation we take x and X Q as new independent variables, then 

dy ^ d (x, y) = 8 (x, y) 3 (x^y^) ^ d_(x^ ,J/Q) 
dx Q d(x, x ) 3 (x, * ) 9 (x, y) 9 (x, x ) ' 



Motion of a Fluid 81 

, dy dy Q 

and so ^- = /. 

OX Q dx 

This means that ydx y Q dx Q is an exact differential and so we may write 



where F = F (x, x , t) and t is regarded as constant. If, however, we allow 
t to vary and use brackets to denote derivatives when x } y and t are 
regarded as independent variables, we have 

dy Q d*F S 2 F 

A "U />/ i 

V 1 ^ 'S 'S I" <* " <> . J 

ratoft OXnCt 



O-i* i- , -^fog^ \Jfo-; > 



1 = 



3y/ " 3a:8x 

_ _ fov 

cte^y 9#9 "^ 9# 
/3a? \ 



= u. 

t\cy / 

O 17f 

Hence we may write = and obtain a convenient expression for the 

stream-function. 

Another physical interpretation of the functions <f> and $ is obtained by 
regarding <f> as the electric potential and u, v as the components of the 
electric field strength due to a set of fictitious point charges, or, if we prefer 
a three-dimensional interpretation, to a system of uniform line charges on 
lines perpendicular to the plane of x, y. The curves cf> = constant are then 
sections by this plane of the equipotential surfaces cf> = constant, while the 
curves = constant are the "lines of force" in the plane of x, y. For 
brevity we shall sometimes think in terms of the fictitious point charges 
and call a curve <f> = constant an "equipotential." 

Again, <f> may be interpreted as a magnetic potential of a system of 
magnetic line charges (fictitious magnetic point charges) or of electric 
currents of uniform intensity flowing along wires of infinite length at right 
angles to the plane of x, y. The curves $ = constant are again lines of force, 
a line of force being defined by the equations 

dx _dy 

u v 



82 The Classical Equations 

In all cases the lines of force are the orthogonal trajectories of the equi- 
potentials, as may be seen immediately from the relation 



dx dx dy dy ~~ ' 

which is a consequence of Cauchy's relations. 

For any number of electric or magnetic line charges perpendicular to 
the plane of x, y we have by definition 

<f> + iif* = 22^ s log [x - x s + i (y - y,)], 

where /x 8 is the density per unit length of the electricity, or magnetism as 
the case may be, on the line which passes through the point (x, y)\ It must 
be understood, of course, that when </> is the electric potential we consider 
only electric charges and when </> is the magnetic potential we consider only 
magnetic charges. When the number of terms in the series is finite we can 

certainly write . . . , . , 

J </ + *0=/(x + ty) =/(), 

where / (z) is a function which is analytic except at the points z = z s . 

When in the foregoing equation p, 8 is regarded as a purely imaginary 
quantity, <f> may be interpreted as the magnetic potential of a system of 
electric currents flowing along wires perpendicular to the plane of x, y. 
If fji s = iC 8 the current along the wire x s , y s is of strength C 8 and flows in 
the positive direction, i.e. the direction associated with the axes Ox, Oy by 
the right-handed screw rule. 

When a potential function <f> is known it is sometimes of interest to 
determine the curves along which the associated force (or velocity) has 
either a constant magnitude or direction. This may be done as follows. We 

have 



log (u - iv) = log/' (x + iy) = <X> + f, say, 
where O = log (u 2 + v 2 ), T = TT tan* 1 (v/u). 

The curves O == constant are clearly curves along which the magnitude 
(u 2 + V 2 )* of the force or velocity is constant, while Y = constant is the 
equation of a curve along which the direction of the force is constant. The 
functions <t> and T are clearly solutions oi Laplace's equations, i.e. 

320 3 2 O _ 

dx 2 + dy 2 ~ 

A function O which satisfies this equation is called a logarithmic 
potential to distinguish it from the ordinary Newtonian potential which 
occurs in the theory of attractions. The electric and magnetic potentials of 
line charges are thus logarithmic potentials. 



Two-Dimensional Stresses 83 

A logarithmic potential <D is said to be regular in a domain D if 



ao ao 

' > ' 2 ' 



are continuous functions of x and y for all points of D. If D is a region 
which extends to infinity it is further stipulated that 

lim <f> (x, y) = (7, lim r - ^ = lim r ~ - = 0, 

r > oo r > oo / > oo 0y 

(r 2 - a: 2 + y*), 

where C is a finite quantity which may be zero. In this sense the potential 
of a single line charge is not regular at infinity. 

Still another physical interpretation of conjugate functions is obtained 
by writing Y y JL y y ,/, 

A x = I y = 0, Ay = I X = </f. 

Cauchy's relations then give 



These are the equations for the equilibrium of an elastic solid when there 
are no body forces and the stress is two-dimensional. The quantities 
(X x , X v ) are interpreted as the component stresses across a plane through 
(x, y) perpendicular to the axis of x, while (Y x , Y v ) are the component 
stresses across a plane perpendicular to the axis of y. The relation X v = Y x 
is quite usual but the relation X x + Y y indicates that the distribution 
of stress is of a special character. A stress system satisfying this condition 
can, however, be obtained by writing 

*-- rv=( 2 - 2 ), Y x = X v =*-uv, 
for these equations give 

SX X dX v /du dv\ /dv du 



The fact that the various potentials <f> and which have been considered 
so far are solutions of Laplace's equation 



is a consequence of the circumstance that they have been defined as sums 
of quantities that are individually solutions of this equation. No physical 
principle has been used except a principle of superposition which states 
that when the individual terms give quantities with a physical meaning, 

6-2 



84 The Classical Equations 

the sum will give a quantity with a similar physical meaning. In the 
analysis of many physical problems such a superposition of individual 
effects is not strictly applicable, for the sources of a disturbance cannot be 
supposed to act independently, each source may, in fact, be modified by 
the presence of the others or may modify the mode of propagation of the 
disturbance produced by another. Such interactions will be left out of 
consideration at present, for our aim is not to formulate at the outset a 
complete theory of physical phenomena but to gradually make the student 
familiar with the mathematical processes which have been used successfully 
in the gradual discovery of the laws of physical phenomena. 

In applied mathematics the student has always found the formulation 
of the fundamental equations of a problem to be a matter of some difficulty. 
Some men have been very successful in formulating simple equations be- 
cause, by a kind of physical instinct, they have known what to neglect. The 
history of mathematical physics shows that in many cases this so-called 
physical instinct is not a safe guide, for terms which have been neglected 
may sometimes determine the mathematical behaviour of the true solution. 
In recent years the tendency has been to try to work with partial differential 
equations and their solutions without the feeling of orthodoxy which is 
created by a derivation of the equations that is regarded for the time being 
as fully satisfactory. The mathematician now feels that it is only by a 
comparison of the inferences from his equations with the results of ex- 
periment and the inferences from slightly modified equations that he can 
ascertain whether his equations are satisfactory or not. In the present 
state of physics the formulation of equations has not the air of finality 
that it had a few' years ago. 

This does not mean, however, that the art of formulating equations 
should be neglected, it means rather that mathematicians should also 
include amongst their special topics of study the processes which lead to 
the most interesting partial differential equations of physics. These pro- 
cesses are of various kinds. Besides the process of elimination from equa- 
tions of the first order there are the methods of the Calculus of Variations 
and methods which depend upon the use of line, surface and volume 
integrals. Mathematically, the direct process of elimination is the simplest 
and will be given further consideration in 1-82. 

/ 

1-72. Geometrical properties of equipotentials and lines of force. When 
the potential </> is a single-valued function of x and y there cannot be more 
than one equipotential curve through a given point P in the (x, y) plane. 
An equipotential curve < = </> may, however, cross itself at a point and 
have a multiple point of any order at a point P Q . In such a case the 
tangents at the multiple point are arranged like the radii from the centre 
to the corners of a regular polygon. To see this, let us take the origin at P , 



Method of Inversion 85 

then the terms of lowest degree in the Taylor expansion of </> < are of 

yP c n e nla - (x 4- iy) n 4- c n e~ nt(l (x iy) n , 

where n is an integer and c and a are constants. In polar co-ordinates 
x = r cos 9, y = r sin #, these terms become 

2c n r n cos ^ (0 4- ), 

and the directions of the n tangents are given by cos n (6 -f ). The possible 
values of n (9 4- a) are thus ?r/2, 37r/2, ... (n J) TT, the angle between con- 
secutive tangents being TT/U. 

Since cos n (9 4- a) is positive for some values of 6 and negative for 
others, the function <f> cannot have a maximum or minimum value at a 
point, for this point may be chosen for origin and the expansion shows that 
there are points in the immediate neighbourhood of the origin for which 
</> > (f) Q , and also points for which < < </> . 

By means of the transformation 
x' - iy' = k* (x 4- iy)- 1 , 
x' -f ^y = & 2 (x - iy)~ l , 
which represents an inversion with 
respect to a circle of radius k and centre 
at the origin, an equipotential curve of 
a system of line charges is transformed 1S ' 

into an equipotential curve of another system of line charges. 

In polar co-ordinates we have 

r' = *Y r > 9 ' = - 
If in Fig. 10 Q corresponds to P and B to A, we have 

Rjr = B'/b, 
where AP = R, OP = r, BQ - R', OB = b. 

For a number of points A and the corresponding points B 

S/t. log (R,/r) = S Ms (log ' - log b). 
An equipotential system of curves represented by the equation 




is thus transformed into an equipotential system represented by the 
equation 



= C _ ^^ log 6. 

A line charge at B is seen to correspond to a line charge of equal strength 
at A and another one of opposite sign at which may be supposed to 
correspond to a line charge at infinity sufficient to compensate the charge 
at B. 

Ap equipotential curve with a multiple point at O inverts into an equi- 
potential which goes to infinity in the directions of the tangents at the 



86 The Classical Equations 

multiple point. This indicates that the directions in which an equipotential 
goes to infinity are parallel to the radii from the centre to the corners of 
a regular polygon. 

In the simple case of two equal line charges at the points (c, 0), ( c, 0) 
the equipotentials are 

log R l + log R 2 = constant, 
or R^ 2 = a 2 , 

where a is constant for each equipotential. These curves are Cassinian 
ovals with the polar equation 

r 4 -f c 4 - 2r 2 c 2 cos 20 = a 4 . 

When a = c we obtain the lemniscate r 2 = c 2 cos 26 with a double 
point at the origin. The tangents at the double point are perpendicular. 

Inverting we get the equipotentials for two equal line charges of 
strength + 1 at the points (6, 0), (- 6, 0), where be = k 2 and a line charge 
of strength 2 at the origin. The equipotentials are now 

log RI + log jR 2 ' - 2 log r' = constant, 
or c*R/jR 2 ' = aV 2 . 

Dropping the primes we have the polar equation 
C 2 ( r 4 + C 4 _ 2r 2 c 2 cos 20) - a*r 4 

of a system of bicircular quartic curves. When a = c we obtain the rect- 
angular hyperbola r 2 cos 20 = c 2 which is the inverse of the lemniscate. 
The rectangular hyperbola goes to infinity in two perpendicular directions. 

It is easily seen that lines of force invert into lines of force. In Fig. 10, 
if we denote the angles POA, PAB, QBO by d, Q and 0' respectively, we 
have the relation _ 

Hence the lines of force represented by the equation 

S fj, s / = constant 
transform into the lines of force represented by the equation 

2/A,0 5 02/i s = constant. 

In particular, the lines of force of two equal line charges 
i H- 2 = constant, 

being rectangular hyperbolas, invert into the family of lemniscates repre- 
sented by _ 



and these are the lines of force of two equal line charges of strength -f 1 
and a single line charge of strength - 2 at 0. 

At a point of equilibrium in a gravitational, electrostatic or magnetic 
field, the first derivatives of the potential vanish and so the equipotential 
curve through the point has a double point or multiple point. A similar 



Lines of Force 87 

remark applies to a curve ifj = constant, but this curve cannot strictly be 
regarded as a single line of force for, if we consider any branch which passes 
through the point of equilibrium without change of direction, the force is 
in different directions on the two sides of the point of equilibrium and the 
neighbouring linqs of force avoid the point of equilibrium by turning through 
large angles in a short distance. This is exemplified in the case of two equal 
masses or charges when the equipotentials are Cassinian ovals which include 
a lemniscate with a double point at the point of equilibrium. The lines of 
force are then rectangular hyperbolas, the system including one pair of 
perpendicular lines which cross at the point of equilibrium. 

In plotting equipotential curves and lines of force for a given system of 
line charges it is very useful to know the position of the points of equi- 
librium, since the properties just mentioned can be employed to indicate 
the behaviour of the lines of force. At a point of stagnation in an irrota- 
tional two-dimensional flow of an inviscid fluid the component velocities 
vanish and so the first derivatives of the velocity potential and stream- 
function are zero. The properties of the equipotentials and stream-lines at 
a point of stagnation are, then, similar to those of equipotential and lines 
of force at a point of equilibrium. There is, however, one important 
difference Between the two cases. In the electric problem the field is often 
bounded by a conductor, i.e. an equipotential surface, while in the hydro- 
dynamical problem the field of flow is generally bounded by some solid 
body whose profile in the plane z = is a stream-line. A point of stagnation 
frequently lies on the boundary of the body and two coincident stream- 
lines may be supposed to meet and divide there, running round the body 
in opposite directions and reuniting at the back of the body when the 
profile is a simple closed curve. 

A point on a conductor may be a point of equilibrium if the conductor's 
profile is a curve with a double point with perpendicular tangents or if it 
consists of two curves cutting one another orthogonally at all their 
common points. It should be remarked, however, that the force at a double 
point may be either zero or infinite ; it is zero when the double point repre- 
sents a pit or dent in the curve, but is infinite when the double point 
represents a peak. This may be exemplified by the equations <f> = x 2 y 2 , 
iff = 2xy. If the field lies in the region x > 0, y 2 < x 2 , the force is zero at 
and there is a single line of force through 0, namely, y = (Fig. 11). 
If, on the other hand, the field is outside the region x < 0, x 2 > y 2 , and 
, _ x 2 -y 2 _ 2xy 



the force at the origin is infinite for most methods of approach and there 
are three lines of force through the origin (Fig. 12). 

Similarly, when two conductors meet at any angle less than ?r, but a 
submultiple of TT, the angle being measured outside the conductor. The 



88 The Classical Equations 

point of intersection is a point of equilibrium and we have the approximate 

< = 2c n r n cos n (6 -f ) 



expression 





of Force 



Line 
of Force 

Fig. 12. 

for the value of </> in the neighbourhood of the point, the equation of the 
conductor in the neighbourhood of the point being n (9 -f ) = n/2 and 

the field being in the region 9 < n (9 -f a) < ~ . The angle is in this case 



7T/n and the radial force ^ varies initially according to the (n l)th power 

of the distance as a point recedes from the position of equilibrium. 
The corresponding approximate expression for /r is 

ifj = 2c n r n sin n (6 + a) 

and there is a single line of force 9 = a which lies within the field, this 
being its equation in the immediate neighbourhood of the point O. 

There is another simple transformation which is sometimes useful for 
deriving the equipotentials and lines of force of one set of line charges from 
those of another. This is the transformation 

z' = z + a 2 /z 
which gives two values of z for each value of z' . Let these be z and z, then 



zz a* 



Similarly, let z 1 and z l correspond to z/, then 



z' - z/ - z - z l -f a*/z - a 2 /*! = (z - zj (1 - 2 

= (Z - 2J (1 - V) = (Z - Z,) (Z - Zj/Z. 

Taking the moduli we obtain a relation 



where 



r = z - z 



r = 



Geometrical Properties 89 

Similarly, if z 2 and z 2 correspond to z 2 ' we have, with a similar notation, 

< = r 2 r 2 fr, 

j r 2 x r 2 r 2 

and so -~ = -~~ . 

r i Wi 

The transformation thus enables us to derive the equipotentials for four 
charges (1, 1, 1, 1) from the equipotentials for two charges (1, 1) 
and a similar remark holds for the lines of force, as may be seen by equating 
the arguments on the two sides of the equation 



This is just one illustration of the advantages of a transformation. 

A general theory of such transformations will be developed in 
Chapter III. 

Some geometrical properties of equipotential curves and lines of force 
may be obtained by using the idea of imaginary points. The pair of points 
with co-ordinates (a ij8, b T ia) are said to be the anti-points of the pair 
with co-ordinates (a a, b /?), the upper or lower sign being taken 
throughout. Denoting the two pairs by F l , F 2 ; S l , S 2 respectively, we can 
say that if S 1 and S 2 are the real foci of an ellipse, then F t and F 2 are the 
imaginary foci. F l and F 2 can also be regarded as the imaginary points of 
intersection of the coaxial system of circles having 8 l and S 2 as limiting 
points. 

If the co-ordinates of F and F 2 are (x l9 yj, (x 2 , y 2 ) respectively arid 
those of S lt S 2 are ( 19 T?^, ( 2 , 7? 2 ) respectively, we have 

x i + iyi = a + * 4- a -f ij8 = & + iifr, 
x \ - iy\ = a ~ ib - a + ip = ^ 2 - i^ 2 , 
x z + iy* = a + ib - - *j3 = ^ 2 -f i^ 2 , 
^2 ~ *2/2 = a i& + a *'j8 = f ! i-^! . 
If now i -f- iv f (x + iy), u iv f (x iy), 

and $! , $ 2 lie on a curve u = constant, we have 



The foregoing relations now show that 

/ te + iyi) + f fa - iy*) = / (* 2 + 

and this means that F l9 F 2 lie on a curve v = constant. 

When the imaginary points on a curve v = constant admit of a simple 
geometrical representation or description, the foregoing result may be 
sometimes used to find the curves u constant. If the curve v = constant 
is a hyperbola, the imaginary points in which a family of parallel lines meet 
the curve have geometrical properties which are sufficiently well known 
to enable us to find the anti-points of each pair of points of intersection. 



90 The Classical Equations 

These anti-points lie on a confocal ellipse which is a curve of the family 
u = constant. By taking lines in different directions the different ellipses 
of the family u constant are obtained. Similarly, by taking a set of 
parallel chords of an ellipse and the anti-points of the two points of inter- 
section of each chord, it turns out that these anti-points all lie on a confocal 
hyperbola, and by taking families of lines with different directions the 
different hyperbolas of the confocal family may be obtained. 

In this case the relations are particularly simple. In the general case 
when one curve of the family u = constant is given there will be, pre- 
sumably, a family of lines whose imaginary intersections with this curve 
are pairs of points with anti-points lying on one curve of the family v = con- 
stant, but these lines cannot be expected, in general, to be parallel, and a 
simple description of the family is wanting. 

EXAMPLES 

1. If a family of circles gives a set of equipotential curves, the circles are either con- 
centric or coaxial. 

2. Equipotentials which form a family of parallel curves must be either straight lines 
or circles. 

[Proofs of these propositions will be found in a paper by P. Franklin, Journ. of Math. 
and Phys. Mass. Inst. of Tech. vol. vi, p. 191 (1927).] 

1-81. The classical partial differential equations for Euclidean space. 
Passing now to the consideration of some partial differential equations in 
which the number of independent variables is greater than two we note 
here that the most important equations are Laplace's equation 

3 2 F 3 2 F 3 2 F 



the wave-equation ^v 9 2 F 9 2 F 1 9 2 F 

"S* + ay + a*' 2 ^c 2 w 

the equation of the conduction of heat 



the equation for the conduction of electricity 

dE 

^ ....... (D) 



and the wave-equation of Schrodinger's theory of wave -mechanics. This 
last equation takes many different forms and we shall mention here only 
the simple form of the equation in which the dependence of on the time 
has already been taken into consideration. The reduced equation is then 



. 
8? + ^ ( ^ = ' ...... ( } 

where F is a function of x, y and z and E is a constant to be determined. 



Laplace's Equation 91 

In these equations K represents the diffusivity or thermometric con- 
ductivity of the medium, K the specific inductive capacity, /x the per- 
'meability, and a the electric conductivity of the medium. The quantities 
c and h are universal constants, c being the velocity of light in vacuum 
and h being Planck's constant which occurs in his theory of radiation. 

Laplace's equation, which for brevity may be written in the form 

V 2 V = 0, 

may be obtained in various ways from a set of linear equations of the first 
order. One set, 

dv dv dv sx dY dz 
x= s*' Y =dy> z = ~dz' a* + ay + al = ' (F) 

occurs naturally in the theory of attractions, V being the gravitational 
potential and X, Y, Z the components of force per unit mass. The last 
equation is then a consequence of Gauss's theorem that the surface integral 
of the normal force is zero for any closed surface not containing any 
attracting matter. 

The same equations occur also in hydrodynamics, the potential V being 
replaced by the velocity potential < and the quantities X, Y, Z by the 
component velocities u, v, w. The equation is then the equation of con- 
tinuity of an incompressible fluid. 

The electric and magnetic interpretations of X , Y, Z and V are similar 
to the gravitational except that the electric (or magnetic) potential is 
usually taken to be V when X , Y , Z are the force intensities. 

As in the two-dimensional theory, Laplace's equation is satisfied by the 
potential V because by the principle of superposition V is expressed as the 
sum of a number of elementary potentials each of which happens to be a 
solution of Laplace's equation, the elementary potential being of type 

V=[(x- x')* +(y- y') 2 + (a - z')T } = l/S. 

When V is interpreted as the electrostatic potential this elementary 
potential is regarded as that of a unit point charge at the point (#', y* ', z') ; 
when V is interpreted as a magnetic potential the elementary potential is 
that of a unit magnetic pole. In the theory of gravitation the elementary 
potential is that of unit mass concentrated at the point (#, y y z). A more 
general expression for a potential is 

V = 2m 5 [(x - *.) + (y - y s )* + (z - .)]-*, 

where the coefficient m s is a measure of the strength of the charge, pole or 
mass concentrated at the point (x 8 , y 8 , z s ). If we write <f> in place of V, 
where <f> is a velocity potential for a fluid motion in three dimensions, the 
elementary potential is that of a source and the coefficient m s can be 
interpreted as the strength of the source at (x 3 , y 8 , z s ). Sources and sinks 



92 The Classical Equations 

are useful in hydrodynamics as they give a convenient representation of 
the disturbance produced by a body when it is placed in a steady stream. 

1-82. Systems of partial differential equations of the first order which 
lead to the classical equations. When we introduce algebraic symbols 

x ~ dx ' v ~~ dy ' z ~~ dz 

for the differential operators the equations (F) of 1-81 become 

X - D x V = 0, 
7 - D y V = 0, 

z - A v = o, 



and the algebraic eliminant 



1 








-A 





1 





-A 








1 


-A 


D 


A 


X) 






- 



is simply Z> x 2 -f Z> v 2 4- A 2 = 0. 

If, on the other hand, we consider the set of equations 

dw dv ds _ 
dy dz dx ' 

du dw ds 

- , _ r= {J 

dz ox dy 

dv du ds _ 

3# 3?/ 82 ~ ' 

du dv dw 

which give V 2 u = V 2 v = V 2 w; = V 2 *s = 0, 

the corresponding algebraic equations 



.(G) 



give the eliminant 



- A M + 


Af 


D v w- 


D x s = 0, 

A* = > 
A = o, 


AM + 


A + 


D z w 


= 





-A 


D y 


-A 


= o, 


A 





-A 


-A 




-A 


-DX 





-A 




A 


/> 


A 








which is equivalent to 



+ 



+ 



(H) 



(I) 



Elastic Equilibrium 93 

These examples show that the problem of finding a set of linear equa- 
tions of the first order which will lead to a given partial differential equation 
of higher order admits a variety of solutions which may be classified by 
noting the power of the complete differential operator (in this case (V 2 ) 2 ) 
which is represented by the algebraic eliminant written in the form of a 
determinant. 

It is known that Laplace's equation also occurs in the theory of elas- 
ticity. If u, v, w denote the components of the displacements and X x , Y y , 
Z z , Y z , Z x , X y the component stresses the equations for the case of no 
body forces are 



.(J) 



* 1 

a# 


v 

ay + 


u^\. z 

Tz'~ 


o, 


dY x 


dYy , 


dY z 


A 


dx 


dy ' 


dz 


U ) 


* 4 


dZ v 


SZ Z _ 


0, 



and if the substance is isotropic the relations between stress and strain 
take the form ~ 



= AA 



.(K) 



where 



V r/ [ ^ i 

7 -- z ' = *(% + &) 



du dw* 

+ dx 

du 



. _ du dv dw 
dx dy dz ' 



The equations obtained by eliminating X x , Y y , Z z , Y z , Z x , X v are 

= (A + ^)~, 



= (A + 11.) ' 



and, except in the case when A + 2/* = 0, a case which is excluded because 
A and fj. are positive constants when the substance is homogeneous, these 



94 The Classical Equations 

equations imply that A is a solution of Laplace's equation. The algebraic 
eliminant is in this case 

(D, 2 + A, 2 + A 2 ) 3 = 0. ...... (L) 

It is easily seen that the quantities X X9 Y y , Z z , 7 C , Z x , X y , u, v, w are 
all solutions of the equation of the fourth order 

V 2 V 2 w = 0, 
i.e. V 4 ^ = 0, 

which may be called the elastic equation. The algebraic equation obtained 
by eliminating the twelve quantities X x , X y , X z , Y X) Y y , Y z , Z x , Z y , Z z , 
u, v, w from the twelve equations (J) and (K) by means of a determinant 
is also equivalent to (L). 

The question naturally arises whether as many as four equations are 
necessary for 'the derivation of Laplace's equation from a set of equations 
of the first order. The answer seems to be yes or no according as we do 
or do not require all the quantities occurring in the linear^quations of the 
first order to be real. Thus, if we write U = u iv, V = w + is, where 
u, v, w, s are the quantities satisfying the equations (H), it is easily seen 

that du .du dv A dv .dv du 

. 1 4 . ___ I __ - I I _ _ n _ A 

~liT~ I V *\ I ?N - V J ~f\ V f* ~ - ~^. ~ - U. 

dx oy dz ex cy dz 

and these equations imply that 

= 0, V 2 F= 0. 



The algebraic eliminant is in this case simply (G). 

It should be noticed that if we write ict in place of y the two-dimensional 
wave-equation 



may be derived from the two equations 

dU dV 13^_ ?Z_^_l?Z_n 

dx + dz + c ~dt ~ ' dx dz c~dt~ 

which have real coefficients. The wave-equation (B) may also be derived 
from two linear equations of the first order 

dU .dU_dV IdV 
dx + l dy ~ ~fa + c dt y 

w__ i dv^i su_du 

dx dy "~ c dt dz ' 

but in this case the coefficients are not all real. The algebraic eliminant is 
in this case simply 

* c 2 (IV + D v 2 -f A 2 ) - A 2 = 0. 



Equations leading to the Wave-Equation 95 

To obtain the wave-equation from a set of linear equations of the first 
order with only real coefficients we may use the set of eight linear equations, 



dy 


dz~ 


'ds 


dx' 


dy 


dz 


~dx 


ds' 


da 


dy_ 


dY 


dT 


dx 


dz 


_dr 


dp 


dz 


dx 


'ds 


dy' 


'dz 


dx 


dy 


ds' 


dp 


da_ 


dZ 


dT 


dY 


dx 


dr 


dy 


dx~ 


dy~ 


ds 


dz' 


'dx 


dy 


~dz 


ds' 


X i 


57_ 


dT 


dz 


da 


dp 


dr 


dy 


Jx + ' 


d~y~ 


ds 


dz' 


dx 4 


'dy 


~ds 


~dz' 



in which for convenience s has been written in place of ct. 

These equations imply that X , Y, Z, T, a, j3, y and r are all solutions 
of the wave-equation. The algebraic eliminant is now 

0* = (D x * + D v * + A 2 - A 2 ) 4 = 0. 

If in the foregoing equations we put T = r = we obtain a set of 
equations very similar to that which occurs in Maxwell's electromagnetic 
theory. The eight equations may be divided into two sets of four and an 
algebraic eliminant. may be obtained by taking three equations from each 
set and eliminating the six quantities X, 7, Z, , /?, y. There are altogether 
sixteen possible eliminants but they are all of type 2 L = 0, where the 
last factor L is obtained by multiplying a term from the first of the two 

D. D v A D. 
D x D y D z D 8 
by a term from the second. 

1-91. Primary solutions. Let/ (&, 2 , ... w ) be a homogeneous poly- 
nomial of the degree in its m arguments & , 2 > w an< ^ ^ e ^ e a ch of the 
quantities D 8 that is used to denote an operator d/dx 8 be treated as an 
algebraic quantity when successive operations are performed. The equation 

/(A,A,-A)*=o (A) 

is then a linear homogeneous partial differential equation of a type which 
frequently occurs in physics. An equation such as 

Dfw = D 2 w 
may be included among equations of the foregoing type by writing 

u = e** . w, 
and noting that u satisfies the equation 

(A 2 ~ A A) u = 0. 
A solution of the form 



96 The Classical Equations 

in which X , 2 , ... 0, are particular functions of.a^, x 2 , x 3 , ... x m , and F is 
an arbitrary function of the parameters 19 2 , 3 , ... >s ., will be called a 
primary solution. An arbitrary function will be understood here to be a 
function which possesses an appropriate number of derivatives which are 
all continuous in some region R. Such a function will be said to be con- 
tinuous (/), n) when derivatives up to order n are specified as continuous. 

It can be shown that the general equation (A) always possesses primary 
solutions of type u = F (0), ...... (B) 

where 6 == c,^ 4- c 2 x 2 -f ... c m x m , ...... (C) 

and c t , c 2 , ... c m are constants satisfying the relation 

/(q,^, ...C TO )-= 0. ...... (D) 

This relation may be satisfied in a variety of ways and when a para- 
metric representation ~ , 



c = 



c ro ~ C'm ( a i> 2> a m-2 

is known for the cD-ordinates of points on the variety whose equation is 
represented by (D), the formulae (B) and (C) will give a family of primary 
solutions. 

When ra = 2 there is generally no family of primary solutions but 
simply a number of types, thus in the case of the equation 

(A 2 - A 2 ) u = o 

there are the two types 

u = F (^ -f a; 2 ), te = J 7 (^ - x 2 ). 

Primary solutions may be generalised by summing or integrating with 
respect to a parameter after multiplication by an arbitrary function of 
the parameter. Thus in the case of Laplace's equation we have a family of 
primary solutions V = F (0) . G (a), where 

= z -f ix cos a -f iy sin , 

and a is an arbitrary parameter. Generalisation by the above method leads 
to a solution which may be further generalised by summation over a 
number of arbitrary functional forms for F (0) and G (a) and we obtain 
Whittaker's solution * 

f2ir 

V = W (z -f ix cos a -f iy sin a, a) da, 
.'o 

which may also be obtained directly by making the arbitrary function F 
a function of a as well as of 0. 

The primary solutions (B) are not the only primary solutions of 

* Math. Ann. vol. LVII, p. 333 (1903); Whittaker and Watson, Modern Analysis, ch. xvin. 



Primary Solutions 97 

Laplace's equation, for it was shown by Jacobi* that if 9 is defined by the 
equation - # (9) + *, (8) + * (9), ...... (F) 

where (6), rj (0) and (0) are functions connected by the relation 

[f W] 2 + h (0)] 2 + K Wl 2 = o, 

then F = .F (8) is a solution of Laplace's equation. 
This is easily verified because iff 

M=l-x? (d) - yr,' (d) - z? (6), 



we have 



These equations give 



M ~ ' [*" (> + OT" w + *r (n - f <> - 



and so V 2 - 0, V 2 {F (0)} - 0. 

This theorem is easily generalised. If c a (a), c 2 (a), ... c w (a) are functions 
connected by the identical relation (D) the quantity 6 defined by the 
equation & = ^ ^ + ^^ (&) + _ ^^ ((?) (Q) 

is such that i/ = ^ (0) is a primary solution of equation (A). 

Since v = du/dx l is also a solution of the same differential equation it 
follows that if G (6) is an arbitrary function and 

M = 1 - * lC / (6) - x 2 c 2 ' (6) - ... x m c m ' (0) 
tne expression v = M~ 1 G (0) 

is a second solution of the differential equation. The reader who is familiar 
with the principles of contour integration will observe that this solution 
may be expressed as a contour integral 

If O (a) da 

V i L 

2-rri ) c a - X& (a) - X 2 c 2 (a) - ... x m c m (a) ' 

where C is a closed contour enclosing that particular root of equation (G) 
which is used as the argument of the function G (0). 

It is easy to verify that the contour integral is a solution of the 
differential equation because the integrand is a primary solution for all 
values of the parameter a and has been generalised by the method already 
suggested. 

* Journal fiir Math. vol. xxxvi, p. 113 (1848); Werke, vol. IT, p. 208. 
f We use primes to denote differentiations with respect to 0. 



98 The Classical Equations 

In this method of generalisation by integration with respect to a para- 
meter the limits of integration are generally taken to be constants or the 
path of integration is taken to be a closed contour in the complex plane. 
It is possible, however, to still obtain a solution of the differential equation 
when the limits of integration are functions of the independent variables 
of type 0. Thus the integral 



re 
V = W (z + ix cos a + iy sin a, a) da 

J o 

V 2 F = when 6 is < 
(a) _ 77 (a) _ (a) 



satisfies Laplace's equation V 2 F = when is defined by an equation of 
type (F) where 



icosa ism a 1 

When the equation (A) possesses primary solutions of type u = F (6) 
and no primary solutions of type u = F (0, <f>) it will be said to be of the 
first grade. When it possesses primary solutions of type u = F (0, <f>) and 
no primary solutions of type u = F (0, <f>, if/) it will be said to be of the 
second grade and so on. 

The equation d 2 u/dxdy = is evidently of the first grade because the 
general solution is u = F (x) -f G (y), where F and G are arbitrary 
functions. The primary solutions are in this case F (x) and G (y). 

Laplace's equation V 2 (u) = is also of the first grade but the equation 

du Su du _ 
dx+dy+dz** 
is of the second grade because the general solution is of type 

u = F (y - z, z - x). 
There is, of course, a primary solution of type 

u= F (y- z,z- x,x- y), 

where F (0, </>, 0) is an arbitrary function of the three arguments 0, <f> y ^r, 
but these arguments are not linearly independent ; indeed, since 

+ < -f ^ = 0, 

a function of 0, <f> and ifi, is also a function of and <f>. In the foregoing 
definition of the grade of the equation it must be understood, then, that 
the parameters 0, <f>, $, etc., are supposed to be functionally independent. 
The differential equation 



has not usually a grade higher than one. If, in particular, an attempt is 
made to find a solution of type 



where = x^ + x 2 & -f z 3 3 -f a? 4 f 4 , 

<f> = #!*?! + #2*72 + *3>?3 + 



Primary Solution of the Wave-Equation 99 

it is found that a number of equations must be satisfied. These equations 
imply that 

where a and b are arbitrary parameters and this means that all points of 
the line 

lie on the surface whose equation is 

/(r r Y v \ 
V^i> ^2) ^3? *'4; ~~ u 

When / (#! , x 2 , x 3 , x 4 ) has a linear factor of the first degree or is itself 
of the first degree the equation (A) is of grade 3. In particular the equation 

possesses the general solution 

Tfl / \ 

and so is of grade 3. An equation with m independent variables which, by 
a simple change of variables, can be written in the form 

a / a a a 



is said to be reducible. Such an equation is evidently of grade m 1. It 
is likely that whenever the number of independent variables is m and the 
grade m 1 the equation is reducible. The wave-equation 



is of grade 2 because there is a primary solution of type 

u = F(0, <), 

where n .... 

x cos a 4- i/ sin a -f tz, q> = x sin a. y cos a 4- 

This solution may be generalised so as to give a solution 

u= ! 'F (0,<f>, a) da, 

Jo 

analogous to Whittaker's solution of Laplace's equation. 



Theequations 



,,_ 

c St ~ ' 



_. 
~dx S^c St 



7-2 



100 The Classical Equations 

which may be written in the abbreviated form * 



and which give the simple equations of Maxwell 



curl 17 =- - 3 j^, div# = 0, 

C vt 



when the vector Q is replaced by H + iE, where E and // are real, may be 
satisfied by writing 



where q (a) is a vector with components cos , sin a, i, respectively and 
F (0, <f>, a) is an arbitrary function of its arguments. 

EXAMPLES 

1. Let , v), , T he functions of a, /9, y connected by the relation 

e + ^ + j 2 + r 2 = i, 

and let X =-- rx - fy + rjz ft + u, Z = yx + (y + TZ & + w, 

Y = & + ry- fz-iit + v, T = (x + ijy + ^2 f- T< + . 
Prove that if the integration extends over a suitable fixed region the definite integral 

V 

satisfies the differential equation 



2. If V = F(A,B,C,D,E) 

is a solution of the equation 

a 2 F a 2 F a 2 F a 2 F_a^F 
a^ 2 + dB* + BC* + en 2 ~ BE* 

when considered as a function of A, B, C, D and E; then, when 

A = 2 (xs -f yw zv tu), G 2 (25 + xv yu tw), 
B = 2 (2/5 4- zu xw tv), D = 2 (^ -f xu + yv -f zw), 

# = a; 2 + y 2 4- z 2 -f ^ 2 -f w 2 + t> 2 + ^ 2 + * 2 , 
the function F is a solution of 

a 2 7 a 2 F a 2 F a 2 F = a 2 F a 2 F &y a 2 F 
a^ 2 + at/ 2 + dz 2 + a* 2 a^ 2 4 " a^ 2 + a^ 2 + a? 

when considered as a function of x, y, z, t, u, v, w, s. 

* We use the symbol Q to denote the vector with components Q x , Qy, Q t respectively. This 
abbreviated form is due to H. Weber and L. Silberstein. 



Characteristics 101 

"'1-92. The partial differential equation of the characteristics. It is easily 
seen that when = c^ -f C 2 x 2 + ... c m x m and c l9 c 2 , ... c m are constants 
satisfying the equation / (c l9 c 2 , ... c m ) = 0, the function F (0) = u is not 
only a primary solution of the equation / (A , Z> 2 , ... Z> m ) w = but it is 
also a solution of the equation 

/(Aw, A*, - AH") = O. (A) 

This partial differential equation of the first order is usually called the 
partial differential equation of the characteristics of the equation 

/(A,A>..-An)* = o. (B) 

In particular, the quantity u = is a solution of this differential 
equation and the locus (x l9 x 2 , ... x m ) = constant is a characteristic or 
characteristic locus of the partial differential equation. 

A characteristic locus can generally be distinguished from other loci of 
type (f> (x l9 x 29 ... x m ) = constant by the property that it is a locus of 
"singularities" or "discontinuities" of some solution of the differential 
equation. If we adopt this definition of a characteristic locus = constant 
it is clear that = constant is a characteristic locus whenever there is a 
solution of the equation which involves in some explicit manner an 
arbitrary function F(0) 9 for the function F(0) can be given a form which 
will make the solution discontinuous on the characteristic locus. 

Thus the quantity u = e~ l l e is a solution of the differential equation (B) 
when = and is discontinuous at each point of the characteristic locus 
0. It should be observed that this function and all its derivatives on 
the side > of the locus 00 are zero for 0=0. The function u = e~ l l b * 
possesses a similar property and the additional one that the derivatives 
on the side < of the locus 0=0 are also zero. From these remarks it 
is evident that if there is a solution of the partial differential equation (B) 
which satisfies the condition that u and its derivatives up to order n 1 
have assigned values on the locus <f> (x ly x 2 , ... x m ) = constant and so gives 
the solution of the problem of Cauchy for the equation, this solution is not 
unique when <f> = because a second solution may be obtained by adding 
to the former one a solution such as e~ 1 / 62 which vanishes and has zero 
derivatives at all points of the locus. This property of a lack of uniqueness 
of the solution of the Cauchy problem for the locus (x l9 x 29 ... x m ) = is 
the one which is usually used to define the characteristic loci of a partial 
differential equation and can be used in the case when the equation does 
not possess primary solutions. Since, however, we are dealing at present 
with equations having primary solutions the simpler definition of as the 
argument of a primary solution or other arbitrary function occurring in a 
solution will serve the purpose quite well. 

An equation with a solution involving an arbitrary function explicitly 
(not under the sign of integration) will be called a basic equation. 



102 The Classical Equations 

Let us now write p 1 = D^u, p 2 = D 2 u, ... so that the partial differential 
equation for the characteristics may be written in the form 

f(Pi>P2> ...Pm) = 0- 
The curves defined by the differential equations 

dx l dx 2 dx m r 

l r== == '"T ...... 

dp l dp 2 dp m 

are called the bicharacteristics * of the equation ; they are the character- 
istics of the equation (A) according to the theory of partial differential 
equations of the first order. 

When p l , p 2 , . . . are eliminated from these equations it is found that 

F (dx l9 dx 2 , ...dx m ) = 0, 

where F (x l , x 2 , . . . x m ) = is the equation reciprocal to/ (p l , p 2 , ... p m ) = 
in the sense of the theory of reciprocal polars. 

In mathematical physics the loci of type u = constant, where u 
satisfies the equation (A), frequently admit of an interesting interpretation 
as wave-surfaces. The curves given by the equations (C) associated with the 
function u are interpreted as the rays associated with the system of wave- 
surfaces. 

In the particular case when the partial differential equation of the 
characteristics is 



d6 36 36 36 36 

#-Si + u * +v t9 + W & 
and u, v and w are constants representing the velocity of a medium and 
V is another constant representing the velocity of propagation of waves 
in the medium, the differential equations of the bicharacteristics are 

dx __ dy __ dz __ dt 

~H5 TTa0 == '" 1 d0 ITs^"""^ ^Tso^Je 
u j* - v V v -j* - v V w ^ - v 5- -J, 

dt ox dt dy dt 4 oz dt 

and the equation obtained by eliminating x- - , ^- , ^- , -^ is 



(dx - udt)* + (dy - vdt) 2 + (dz - wdt)* = 

This result is of considerable interest in the theory of sound and may be 
extended so as to be applicable to the case in which u, v, w and V are 
functions of x, y, z and t. 

It may be remarked that if we have a solution of (D) in the form of a 

complete integral . , m 

r & = t - T - g (x, y, z, a, j3), 

* See J. Hadamard's Propagation des Ondes. The theory is illustrated by the analysis of 19. 



Bicharacteristics and Rays 103 

in which r, a and j3 are arbitrary constants, the rays may be obtained by 
combining the foregoing equation with the equations 

3 9 _ A ^ - o 
a- U ' 3)8 ~" U ' 

The characteristics of a set of linear equations of the first order may be 
defined to be the characteristics of the partial differential equation obtained 
by eliminating all the dependent variables except one. The relation of the 
primary solutions of this equation to the dependent variables in the set of 
equations of the first order is a question of some interest which will now be 
examined. 

Let us first consider the equations 

du __dv du _ dv 

dx~~dy' dy^fa' ...... W 

which lead to the equation 



In this case the quantity w = u + v satisfies a linear equation of the 

first order ~ ~ 

cw _ dw 

fa^dy' 

and this equation possesses the primary solution w = F (x -f y) which is 
also a primary solution of the equation (F). 

Similarly the quantity z = u v satisfies the equation 



which possesses a primary solution z = G (x y) which is also a primary 
solution of the equation (F). 

To generalise this result we consider a set of m linear partial differential 
equations of the first order, 

L^UI + L l2 u 2 -f ... L lm u m - 0, j 
L 21 U} -f L 22 u z -f ... L 2m u m = 0, 1 

L ml Ui + L m2 u 2 + ... L mm u m = Oj 
where L vq denotes a linear operator of type 

(P> V> l ) A + (P> V> 2) D 2 + ... (p, q, m) D m , 

where the coefficients (p, q, r) are constants. 

Multiplying these equations by coefficients b l9 b 2) ... b m respectively, 
the resulting equation is of the form 

L (a^ -f a 2 u 2 + ... a m u m ) = ...... (H) 



104 The Classical Equations 

if the constants b lt b 2 , ... 6 m ; a l9 a 2 , ... a m are of such a nature that 



4- 



a) 



and the operator L is of the form 



where the operator coefficients Z l5 2 , ... / w are constants to be determined. 
Equating the coefficients of the operator D in the identities (I) we 

obtain _,, , v 7 

S6 p (p, ?;r) = a a J f . 

This equation indicates that if z 1) z 2 , ... z m ; y l , t/ 2 > - 2/m are arbitrary 
quantities, the bilinear form 



can be resolved into linear factors 



When the coefficients b ly ... b m can be chosen so that the bilinear form 
breaks up in this way the two factors will give the required coefficients 
a l9 ... a m ; 1 19 ... l m and the partial differential equations will give an ex- 
pression for (Zji/j -f ... a m u m which may be called a primary solution of the 
set of linear partial differential equations of grade m ~ 1. When such a 
solution exists the system is said to be reducible. The problem of finding 
when a set of equations is reducible is thus reduced to an algebraic problem. 

Now let f2 denote the determinant 



Anl An2 Anro 

and let A n , ... A lTO denote the co-factors of the constituents L n , L 12 , ... L lm 
respectively. If we write 

it is easily seen that the last m 1 equations of the set are all formally 
satisfied, and since 

the first equation is formally satisfied if v is a solution of the partial 
differential equation 

which is of order m. Since 

}&! = flA u v = A u lv 0, 

the quantities Ui,u 2 , ... u m are all solutions of the same partial differential 
equation. 



= 0, 



Eeducibility of Equations 105 

It should be noticed that 

a^ + ... a m u m = (a 1 A n + a 2 A 12 + ... a m A lm ) v, 
consequently 

L (fl^Wj + ... a m u m ) = L (a 1 A n -f a 2 A 12 + ... a m A lm ) v. 

The equation (a^ + ... a m u m ) = will be a consequence of the equation 
lv if the operator Cl breaks up into two factors L and 

(a x A n + a 2 A 12 + ... a m A lm ) 

of which one, L, is linear. The set of linear equations is thus reducible when 
the equation lv = is reducible. 

It is clear from this result that we cannot generally expect a set of linear 
homogeneous equations of type (G) to possess primary solutions of grade 
m 1. 

The equations do, however, generally possess primary solutions of 
grade 1. To see this we try 

*i=/i(0), 2=/2(0), - *m=/*(0). 

Substituting in the set of equations we obtain the set of linear equations 
// (0) L n + / 2 ' (0) L 12 + ... f m ' (0) L lm 6 = 0, 
//'() V +/' (0) L 22 + .../ m ' (0) L 2m = 0, 



from which the quantities// (0), f 2 (0), ... f m ' (0) may be eliminated. The 

resulting equation, T a T a T a 

L n U L 12 V ... L lm u 

L 21 L 22 ... L 2m 

. L ml L m2 ... L mm 

is no other than the partial differential equation of the characteristics of 
the equation lu = 0. 

1-93. Primary solutions of the second grade. We have already seen 
that the wave-equation possesses primary solutions of type F (0 y <f>) which 
may be called primary solutions of the second grade. The result already 
obtained may be generalised by saying that if 1 , m , n , p , 1 19 m l9 n l9 p l 
are quantities independent of #, t/, z and t and connected by the relations 

the quantities 

are such that the function u = F (0, <f>) is a solution of Q 2 u = 0. 



106 The Classical Equations 

This result may be generalised still further by making the coefficients 
Z , m , etc., functions of two parameters a, r and forming the double con- 
tour integral 

tt= _ Jiff _ f(a,T)dadT _ 
47T 2 JJ (I x 



_ 

(I x + m y + n Q z - p Q ct - g ) (^x -f 

where/ (a, T), gr (a, r), ft (a, r) are arbitrary functions of their arguments. 

This integral will generally be a solution of the wave-equation and the 

value of the integral which is suggested by the theory of the residues of 

double integrals is T , . . n . 

* u = J-*f(a,P), 

in which a, /? satisfy 

M>j8) = 0, Ao(a,j8) = 0, ...... (K) 

where 

A! (a, r) - #/! (a, r) + ym l (a, r) + zn^ (a, r) - Ctp 1 (a, r) - ft (a, T), 
A (<r, r) = xl (a, r) + ym Q (a, r) -f 2W (a, r) - Ctp (a, r) - gr (a, r), 
and J is the value when cr = a, r = ^8 of the Jacobian 



This result, which may be extended to any linear equation with a two- 
parameter family of primary solutions of the second grade, will now -be 
verified for the case of the wave-equation. It should be remarked that the 
method gives us a solution of the wave-equation of type 



where y is a particular solution of the wave-equation. Such a solution will 
be called a primitive solution ; it is easily verified that the parameters a and 
j3 occurring in a primitive solution are such that the function v = F (a, j8) 
is a solution of the partial differential equation of the characteristics 



Instead of considering the wave-equation it is more advantageous to 
consider the set of partial differential equations comprised in the vector 
equations . ^ 

=0 ...... (M) 



and to look for a primitive solution of these equations of type 



in which / is an arbitrary function of the two parameters a and /?, which 
are certain functions of x, y, z and t, and the vector q is a particular solution 
of the set of equations. 

Substituting in the equations (L) we find that since /is arbitrary a and 
]8 must satisfy the equations in o>, 

cVco x q = iqda/dt, <?.Vaj = 0, 



Primitive Solution of Maxwell's Equations 107 

which indicate that 



^ 3 (.)_** 8 (,) 
- K d(y,z)~ c 3(x,t)' 

_ a (a, )8) uc 9 (a. )8) 
"3(2,*) c8(y, 0' 

8 * 9 (<*. 



(N) 



where K is some multiplier. To solve these equations we take a, /?, #, y as 
new independent variables and write the equation connecting a and ft in 

the form , ~ JX ^ / o ^ 

8 (g, ft, x, t) = * 8 (a, ft, y, g) 

3 (y, z, x,t) cd (x, t, y, z) ' 



_ 
(z, x, y,t) c d (y, t, z, x) ' 



_ ^ 

9 (a:, y, g, ) c 3 (z, t, x, y) ' 

Now multiply each of the Jacobians by ~-? J -%-* \ and make use of 
* J J a(a,j3,,y) 

the multiplication theorem for Jacobians. We then obtain a set of equations 
similar to the above but with 3 (a, /J, x, y) in each denominator. The new 
equations reduce to the form 

dt __ i dz dt _ i dz d (z, t) _ i 
dy cdx' dx ~~ c dy ' 8 (x, y) c ' 

The first two of these equations are analogous to the equations con- 
necting conjugate functions t and ig/c, consequently we may write 

z-ct = &[x+ iy,a, /?], 
z + ct = g [x 4y* a, j8] . 

Substituting in the third equation, we find that 

$'' = - 1, 

where in each case the prime denotes a derivative with respect to the first 
argument. Evidently &' must be independent of x -f iy and <' inde- 
pendent of x iy. The general solution is thus determined by equations 
of the form 8 _ d _ ^ (a , + (jr + iy) d (, ft, 

Z + ct = 4,(a,p)-(x- iy) [0 (a, /3)]- 1 , 

where 0, <j>, ifi are arbitrary functions of a and /9 which are continuous (Z), 1) 
in some domain of the complex variables a and ft. 

For some purposes it is more convenient to write the equations in the 

equivalent form , , / rt . / . v n , /,, 

H z - ct = < (a, j8) + (x + ty) 6 (a, /5), 

(a, ft (z + c<) = X (, 0) - (* - * 



108 The Classical Equations 

These equations are easily seen to be of the type (K) and may indeed 
be regarded as a canonical form of (K). When the expressions for q are 
substituted in the equations (M) it is easily seen that K is a function 
of a and ft. Since Q already contains an arbitrary function of a and ft we 
may without loss of generality take K = 1. 

A case of particular interest arises when 

< = (a) - cr (a) - [f (a) + iy (a)] 0, 
= (a) + cr (a) + [ (a) - irj (a)] 6~\ 

where f (a), 77 (a), () and r () are real arbitrary functions of a which are 
continuous (D, 2). We then have 

^TI^"T[^^i^)j ^ ~ r~~ Ti^+7ir-~T(a)] ' 

and so a is defined by the equation 

We may without loss of much generality take r (a) = a and use r as 
variable in place of a. Let us now regard (T), 77 (T), (T) as the co- 
ordinates of a point S moving with velocity v which is a function of r. 
For the sake of simplicity we shall suppose that for each value of r we 
have the inequality v 2 < c 2 , which means that the velocity of S is always 
less than the velocity of light. We shall further introduce the inequality 
T < t. This is done to make the value of r associated with a given space- 
time point (x, y, z, t) unique*. 

To prove that it is unique we describe a sphere of radius c (t r) with 
its centre at the point occupied by S at the instant r. As r varies we 
obtain a family of spheres ranging from the point sphere corresponding 
to r = t to a sphere of infinite radius corresponding to r = oo. 

Now, since v* < c 2 it is easily seen that no two spheres intersect. Each 
sphere is, in fact, completely surrounded by all the spheres that correspond 
to earlier times r. There is consequently only one sphere through each 
point of space and so the value of r corresponding to (x, y, z, t) is unique. 
The corresponding position of 8 may be called the effective position of S 
relative to (x, y, z, t). 

In calculating the Jacobians r may be treated as constant in the 
differentiations of ft. Now 



* Proofs of this theorem have been given by A. Li6nard, L'&lairage tiectrique, t. xvi, pp. 5, i ^, 
106 (1898); A. W. Con way, Proc. London Math. Soc. (2), vol. I (1903); G. A. Schott, Electromagnetic 
Radiation (Cambridge, 1912). 



Primitive Solution of the Wave-Equation 109 

where 

M=[x- (r)] '(T) + (y ~ V ( T )] VW + [Z - I (r)] '(T) - f a (' - T), 
and primes denote derivatives with respect to T. We thus find that 

'(y, 2) = ~2J/ (1 ~^ )> 

a (, 0) /s 



a(3,y) JIT 

The ratios of the Jacobians thus depend only on ft and we have the general 
result that the function M-I f / R\ 

is a solution of the wave-equation. 

When the point (, T?, ) is stationary and at the origin of co-ordinates 
this result tells us that if,/ is an arbitrary function which is continuous 
(/), 2) in some domain of the variables a, ft and if r 2 = x* + ?/ 2 -I- z 2 the 

function , 

z - r 

* + ?y 

is a solution of the wave-equation. There is a corresponding primitive 
solution of type 



obtained by changing the sign of t and using another arbitrary function. 

In the case of the wave-function M~ l f (a, /J) the parameter a may be 
called a phase-parameter because it determines the phase of a disturbance 
which reaches the point (x, y, z) at time t when the function / is periodic 
in a. The parameter ft is on the other hand a ray-parameter because a 
given complex value of /? determines the direction of a ray when a is given. 

It is easily deduced from the equations (N) that and /3 satisfy the 
differential equation of the characteristics 



, 

and that + 



It follows that the quantity v = F (a, ft) is also a solution of (L). 

An interesting property of this equation (0) is that if a is any solution 
and we depart from the space-time point (x, y, z, t) in a direction and 
velocity defined by the equations 



dx dy dz - r ] sav 

_ = _ = - = - -- as, say, 

da ca da da 

dx dy dz dt 



110 The Classical Equations 

then a and its first derivatives are unaltered in value as we follow the 
moving point. We have in fact 

, da 7 da , , da , da 7j 
da=^dx+* r ay+~ dz + ^ at 

ox oy * oz ot 






2 2 



3% , 3 2 a , r 3 2 

+ 33" ^ + 



[3a 3 2 
3i fa 



3 2 3 3 2 3a 3 2 a 1 da , 

3~z 3iaz " c 2 



- 0. 
Also, if a and /J are connected by an equation of type (P), 



3a 36 1 3a 3B\ 7 

, .i_ ^_ . " ) /7o 

dz dz c 2 dt dtj 
= 0. 

The equations (0 and P) thus indicate that the path of the particle 
which moves in accordance with these equations is a straight line described 
with uniform velocity c and is, moreover, a ray for which j3 is constant. 

1*94. Primitive solutions of Laplace's equation. As a particular case 
of the above theorem we have the result that the function 



r \x 4- iy/ 

is a primitive solution of Laplace's equation. This is not the only type of 
primitive solution, for the following theorem has been proved*. 

In order that Laplace's equation may be satisfied by an expression of 
the form V = yf (0), in which the function / is arbitrary, the quantity 6 
must either be defined by an equation of the form 

[^ - (6)1* +(y~r, (*)] + [z - C (0)? = o, 
or by an equation of the form 

xl (6) + ym (6) + zn (6) = p (6), 
where /, m, n are either constants or functions of 6 connected by the re- 

lation P + m + n-0. 

The most general value of y is in each case of the form 
y = no (6) + y*b (6), 

* See my Differential Equations, p. 202. 



Primitive Solutions of Laplace's Equation 111 

where y x and y 2 are particular values of y, whose ratio is not simply a 
function of 9. In the first case we may take 

n = w-t, 72 = %-*, 

where u; = [a; - (0)] A (0) + [y - rj (6)] ^ (0) + [2 - (0) ] (0), 
t*i = [s - (#)] \ (0) + [y - rj (6)] ^ (6) + [z - (6)] v, (6), 

and A, ^, v, A x , /z x , v l are two independent sets of three functions of 6 which 
satisfy relations of type 

A 2 + ^ + v 2 = 0, 

A (0) r (0) + p, (0) r)' (6) + v (6) ' (0) = 0. 
In the second case we may take y l ~ 1 and define y 2 by the equation 

yr i = ^ (fl) + ym' (9) + zri (9) - p' (9). 
If in the first theorem we choose = 0, rj i9, ~ 0, we have 

r 2 

9=- ----- .--, A-f iu= 0, v = 0, 

x + ly ^ 

iv = x + iy, 
and the theorem tells us that the function 



is a primitive solution of Laplace's equation. If we write x + iy = t, 
x iy = 4:8 this theorem tells us that the function 

F=r*/(4* + z 2 /*) 
is a primitive solution of the equation 

d*V 



_ 

dz* ~ 

1-95. Fundamental solutions*. The equations with primary and 
primitive solutions have been called basic because it is believed that 
solutions of a differential equation with the same characteristics as a basic 
equation can be derived from solutions of the basic equation by some 
process of integration or summation in which singularities of these solutions 
of the basic equation fill the whole of the domain under consideration. 

This point will be illustrated by a consideration of Laplace's equation 
as our basic equation. 

We have seen that there is a primitive solution of type 



r * \x + 
By a suitable choice of the function / we obtain a primitive solution 

* These are also called elementary solutions. See Hadamard, Propagation des Ondes. 



112 The Classical Equations 

with singularities at isolated points and along isolated straight lines issuing 
from isolated singular points. The particular solution 

F= 1/r 

has the single isolated point singularity x = 0, y = 0, z = 0. Let us take 
this particular solution as the starting-point and generalise it by forming 
a volume integral 



over a portion of space which we shall call the domain &). 

When the point (x, y, z) is in the domain I/ 1 this integral is riot a solution 
of Laplace's equation but is generally a solution of the equation 

V 2 V + *vF(x, y, 3) = 0, ...... (B) 

provided suitable limitations are imposed upon the function F. 

Now the function F is at our disposal and in most cases it can be chosen 
so as to represent the terms which make the given differential equation 
differ from the basic equation of Laplace. It is true that this choice of F 
does not give us a formula for the solution of the given equation but gives 
us instead an integro-differential equation for the determination of the 
solution. Yet the point is that when this equation has been solved the 
desired solution is expressed by means of the formula (A) in terms of 
primitive solutions of the basic equation. 

A solution of the basic equation which gives by means of an integral a 
solution of the corresponding equation, such as (B), in which the additional 
term is an arbitrary function of the independent variables, is called a 
fundamental solution. Rules for finding fundamental solutions have been 
given by Fredholm and Zeilon. In some cases the solution which is called 
fundamental seems to be unique and the theory is simple. In other cases 
difficulties arise. In any case much depends upon the domain W and the 
supplementary conditions that are imposed upon the solution. 

When the basic equation is the wave-equation the question of a funda- 
mental solution is particularly interesting. There are, indeed, two solutions, 



J7 1 I 1 1 

and F= - .-}- 



2r [r-ct ' r + ct] r* - cW 
which may be regarded as natural generalisations of the fundamental 
solution 1/r of Laplace's equation. The former seems to be the most useful 
as is shown by a famous theorem due to Kirchhoff . 

In the case of the equation of the conduction of heat the solution which 
is regarded as fundamental is 



Fundamental Solutions 113 

when the equation is taken in the form 



_ X* 

and is V = t~^ e *** 

when the equation is taken in the simpler form 



The equation of heat conduction is not a basic equation but may be 
transformed into a basic equation by the introduction of an auxiliary 
variable in a manner already mentioned. Thus the basic equation derived 
from 97 



w 

is =-3- = -s- 9 - , W = 

3s 3J dx 2 

and this equation possesses the primitive solution 



of which TF = 2~i exp - 

r \_K 4:Ktj 

is a special case. 

The theory of fundamental solutions is evidently closely connected 
with the theory of primitive solutions but some principles are needed to 
guide us in the choice of the particular primitive solution which is to be 
regarded as fundamental. The necessary principles are given by some 
general theorems relating to the transformation of integrals which are 
forms or developments of the well-known theorems of Green and Gauss. 
These theorems will be discussed in Chapter II. An entirely different 
discussion of the fundamental solutions of partial differential equations 
with constant coefficients has been given recently by G. Herglotz, Leipzig er 
Berichte, vol. LXXVIII, pp. 93, 287 (1926) with references to the literature. 

EXAMPLES 

1. Prove that the equation -. = ^-j 

ot dxr 

is satisfied by the two definite integrals 

V - 4 f e-** (cos xs - sin xs) e'* t8 ' da, 

/GO 

F=*/ v(a,t)v(x,8)ds, 



where v (x, t) 

Show also that the two integrals represent the same solution. 



114 The Classical Equations 

2. Prove that this solution can be expanded in the form 

V - F - F, + F, 

where F = r (i) (4<F l [l - 4, fj + ^ (5)" + ) 



3. Show also that 

y = 4 / e~ 4< * 4 cos sx cosh sxds, 
7o 

V l 4 I e~* i8 * [sin sx cosh sx + cos sx sinh 50;] e&, 
Fo == 4 I e~ 4<s4 sin sx sinh so; . ds. 

2 Jo 

4. Prove that there is a fourth solution 

V 3 - x?1r l oi ~ 7 i T "^" n i"\l ( < ) ~ " " ^ / e ~ 4<s4 t sm 5a: C08 ^ 5a: cos sx smn 5a: ] ^ 5 - 

5. If V (x, t) is a solution of the equation 

3V __ d*V 

~df " to* * ~ lj 
the quantity 



is generally a solution of the set of equations 

In particular, if s =. 2 and V (x, t) is the function v (x, t) of Ex. 1, the corresponding 
function y n (t) is 



y n (0 



This may be called the fundamental solution, and when the second form is adopted 
s may have any positive integral value. In particular, when 5 = 4, this function is de- 
rivable from t^e function v (x, t) of Ex. 1, p. 113. 



CHAPTER II 

APPLICATIONS OF THE INTEGRAL THEOREMS 
OF GAUSS AND STOKES 

. In the following investigations much use will be made of the 
well-known formulae 



...... (A) 

for the transformation of line and surface integrals into surface and volume 
integrals respectively. In these equations Z, m, n are the direction cosines 
of the normal to the surface element dS, the normal being drawn in a 
direction away from the region over which the volume integral is taken or 
in a direction which is associated with the direction of integration round 
the closed curve C by the right-handed screw rule. 

The functions u, v, w, X, Y, Z occurring in these equations will be 
supposed to be continuous over the domains under consideration and to 
possess continuous first derivatives of the types required* . The equations 
may be given various vector forms, the simplest being those in which 
u, v, w are regarded as the components of a vector q and X, Y, Z the 
components of a vector F. The equations are then 

I q . ds = I (curl q) . dS, 

j _ J 

J F. dS = [(div F) dr (dr = dx . dy . dz), 

^ 
where ds now stands for a vector of magnitude ds and the direction of the 

tangent to the curve C, while dS represents a vector of magnitude dS and 
the direction of the outward-drawn normal. The dot is used to indicate a 
scalar product of two vectors. Another convenient notation is 

\qtds = I (curlq) n dS, 

J ~ ...... (C) 



/ F n dS f(divF)dr, 



* See for instance Goursat-Hedrick, Mathematical Analysis, vol. I, pp. 262, 309. Some in- 
teresting remarks relating to the proofs of the theorems will be found in a paper by J. Carr, Ph il. 
Mag. (7), vol. iv, p. 449 (1927). The first theorem is well discussed by W. H. Young, Proc. London 
Math. Soc. (2), vol. xxiv, p. 21 (1926); and by O. D. Kellogg, Foundations of Potential Theory, 
Springer, Berlin (1929), ch. iv. 



8-2 



116 Applications of the Integral Theorems of Gauss and Stokes 

where the suffixes t and n are used to denote components in the direction 
of the tangent and normal respectively. 

If we write Z - v, Y = - w, X = 0; X = w, Z = - u, Y = 0; 7 = ?/, 
X = v, Z = in succession we obtain three equations which may be 
written in the vector form 



I (qx dS) = - j(curlq)dr, (D) 



where the symbol x is used to denote a vector product. 

Again, if we write successively X = p, Y = Z = 0; Y = p, Z = X = 0; 
Z = p, X = Y = 0, we obtain three equations which may be written in 
the vector form ^ 

...... (E) 



where Vp denotes the vector with components ^ , ^ ^ , ~ respectively. 

2-12. To obtain physical interpretations of these equations we shall 
first of all regard u, v, w as the component velocities of a particle of fluid 
which happens to be at the point (#, y, z) at time t. The quantities , 77, 
defined by the equations 

. _ dw dv _du dw ^ _ Sv du 
*=dy~dz' r] ~dz~dx 9 ^^dx^dy 

may then be regarded as the components of the vorticity. 

The line integral in (A) is called the circulation round the closed curve 
C and the theorem tells us that this is equal to the surface integral of the 
normal component of the vorticity. When there is a velocity potential 
</> we have ~ , - , ~ , 

dd> O(h o<p 

u = -f- , v = -- , w = /- 
dx dy dz 

(in vector notation q Vcf>) and = y = = 0, the circulation round a 
closed curve is then x zero so long as the conditions for the transformation 
of the line integral into a surface integral are fulfilled. The circulation is 

not zero when . , . , , x 

</> = tan- 1 (y/x), 

and the curve C is a simple closed curve through which the axis of z passes 
once without any intersection. The axis of z is then a line of singularities 
for the functions u and v. The value of the integral is 27r, for ^ increases by 
2n in one circuit round the axis of z. The velocity potential 

</> = (r/27r) tan-* (y/x) 

may be regarded as that of a simple line vortex along the axis of z, the 
strength of the vortex being represented by the quantity F which is- 
supposed to be constant. F represents the circulation round a closed curve 
which goes once round the line vortex. 



Equation of Continuity 1F7 

If we write X = pu, Y = pv, Z = pw, where p is the density of the fluid, 
the surface integral in (A) may be interpreted as the rate at which the 
mass of the fluid within the closed surface S is decreasing on account of 
the flow across the surface 8. If fluid is neither created nor destroyed 
within the surface this decrease of mass is also represented by 



The two expressions are equal when the following equation is satisfied at 
each place (#, y, z) and at each time t, 



This is the equation of continuity of hydrodynamics. There is a similar 
equation in the theory of electricity when p is interpreted as the density 
of electricity and u, v, w as the component velocities of the electricity 
which happens to be at the point (#, ?/, z) at time t. When p is constant 
the equation of continuity takes the simple form 

du dv div __ 

dx + dy + dz = 

(in vector notation div q = 0). This simple form may be used also when 
dp/dt = 0, where d/dt stands for the hydrodynamical operator 

d a a a a 

i4 = ^ + u n + v ~T + w a~ 
cM ra cty dz 

a fluid for which rfp/Y/ = is said to be incompressible. 

When p is interpreted as fluid pressure the equation (E) indicates that 
as far as the components of the total force are concerned the effect of fluid 
pressure on a surface is the same as that of a body force which acts at the 
point (x 9 y, z) and is represented in magnitude and direction by the vector 
Vp, the sign being negative because the force acts inwards and not 
outwards relative to each surface element. Putting q = pr in equation (D), 
where r is the vector with components x, y, z, we have an equation 

I (r x pds) = I (curler) dr = (r x Vp) dr, 

which indicates that the foregoing distribution of body force gives the 
same moments about the three axes of co-ordinates as the set of forces 
arising from the pressures on the surface S. The body forces are thus 
completely equivalent to the forces arising from the pressures on the 
surface elements. This result is useful for the formulation of the equations 
of hydrodynamics which are usually understood to mean that the mass 
multiplied by the acceleration of each fluid element is equal to the total 
body force. If in addition to the body force arising from the pressure there 
is a body force F whose components per unit mass are JL , 7, Z for a particle 



118 Applications of the Integral Theorems of Gauss and Stokes 

which is at (x, y, z) at time t, the equations of hydrodynamics may be 
written in the vector form 



When viscosity and turbulence are neglected the body force often can be 
derived from a potential fi so that F = VQ. The hydrodynamical equations 
then take the simple form , 



which implies that in this case there is an acceleration potential if p is a 
constant or a function of p. When in addition there is a velocity potential 
<f> the equations may be written in the form 



and imply that f dp ~ + ?J + | 9 2 = Q + / (t), 

J p 01 

where/ (t) is some function of t. This may be regarded as an equation for 
the pressure, when ti = it indicates that the pressure is low where the 
velocity is high. 

2-13. The equation of the conduction of heat. When different parts of 
a body are at different temperatures, energy in the form of heat flows from 
the hotter parts to the colder and a state of equilibrium is gradually 
established in which the temperature is uniformly constant throughout 
the body, if the different parts of the body are relatively at rest and do 
not participate in an unequal manner in heat exchanges with other bodies. 
When, however, a steady supply of heat is maintained at some place in the 
body, the steady state which is gradually approached may be one in which 
the temperature varies from point to point but remains constant at each 
point. 

A hot body is not like a pendulum swinging in air and performing a 
series of damped oscillations as the position of equilibrium is approached, 
it is more like a pendulum moving in a very viscous fluid and approaching 
its position of equilibrium from one side only. The steady state appears, 
in fact, to be approached without oscillation. 

These remarks apply, of course, to the phenomenon of conduction of 
heat when there is no relative motion (on a large scale) of different parts 
of the body. When a liquid is heated, a state of uniform temperature is 
produced largely by convection currents in which part of the fluid Amoves 
from one place to another and carries heat with it. There are convection 
currents also in the atmosphere and these are responsible not only for the 
diffusion of heat and water vapour but also for a transportation of momentum 
which is responsible for the diurnal variation of wind velocity and other 
phenomena. 



Conduction of Heat 119 

A third process by which heat may be lost or gained by a body is by 
the emission or absorption of radiation. This process will be treated here 
as a surface phenomenon so that the laws of emission and absorption are 
expressed as boundary conditions ; the propagation of the radiation in the 
intervening space between two bodies or between different parts of the 
same body is considered in electromagnetic theory. The mechanism of the 
emission or absorption is not fully understood and is best described by 
means of the quantum theory and the theory of the electron. The use of a 
simple boundary condition avoids all the difficulty and is sufficiently 
accurate for most mathematical investigations. In many problems, how- 
ever, radiation need not be taken into consideration at all. 

The fundamental hypothesis on which the mathematical theory of the 
conduction of heat is based is that the rate of transfer of heat across a 
small element dS of a surface of constant temperature (i.e. an isothermal 
surface) is represented by 



where K is the thermal conductivity of the substance, is the temperature 
in the neighbourhood of dS, and ~~ denotes a differentiation along the 

outward-drawn normal to dS. The negative sign in this expression simply 
expresses the fact that the flow of heat is from places of higher to places 
of lower temperature. The rate of transfer of heat across any surface 
element dv in time dt may be denoted by f v dadt, where the quantity f v is 
called the flux of heat across the element and the suffix v is used to indicate 
the direction of the normal to the element. 

Let us now consider a small tetrahedron DABC whose faces DBC, 
DC A, DAB, ABC are normal respectively to the directions Ox, Oy, Oz, Ow, 
where the first three lines are parallel to the axes of co-ordinates. Denoting 
the area ABC by A, the areas DBC, DC A, DAB are respectively w x &, 
Wy&, w z A, where w x , w y , w z are the direction cosines of Ow. 

Wher A is very small the rate at which heat is being gained by the 
tetrahedron at time t is approximately 

(w x f x + w v f y + wj z - /) A. 

7/J 

This must be equal to. Vcp ^~ , where V is the volume of the tetrahedron, 

u/t 

c the specific heat of the material and p its density. Now V = jpA, where 
p is the perpendicular distance of D from the plane ABC, hence 

Wxfx + Wyfv + Wzfz - / = $PCP ^ 

and so tends to zero as p tends to zero. 

When DAB is an element of an isothermal surface we may use the 



120 Applications of the Integral Theorems of Gauss and Stokes 
additional hypothesis that f x and f y are both zero and the equation 

giV6S f f 

L, -= w z j z = 

Jw zjz 

The law (A) thus holds not simply for an isothermal surface but for 
any surface separating two portions of the same material. The vector A0 

whose components are x- - , ~ , ~ is called the temperature gradient at the 

point (x, y, z) at time t. 

Let us now consider a portion of the body bounded by a closed surface 
R. Assuming that f ff , f y , f z and their partial derivatives with respect to 
x, y and z are continuous functions of x, y and z for all points of the region 
bounded by /V, the rate at which this region is gaining heat on account of 
the fluxes across its surface elements is 



Transforming this into a volume integral and equating the result to 

J7/3 
cp ' dxdydz, 
(IT 

we have the equation 

! \ C P n i -5~ ( & * ' " -r K ^~ > ' ~>- ( ^ ^ } \ dxdydz. 
]jj[ r dt 3x{ 3x 3y 3y 3z\ 3z J y 

This must hold for any portion of the material that is bounded by a simple 
closed surface and this condition is satisfied if at each point 

cp ( T ._ div (KV0) = 0. 
a I 

If the body is at rest we can write ^ in place of -j- , but if it is a moving 

ut (Jit/ 

Jf\ 

fluid the appropriate expression for -7- is 

d9 30 30 30 30 

~n ^ bi 'I ^ ^ f- V >r -h W x- , 

dt dt ox cy 3z 
where u, v and w are the component velocities of the medium. 

In most mathematical investigations the medium is stationary and the 
quantities c, K and p are constant in both space and time and the equation 
takes the simple form ^ 



dt 

in which K is a constant called the diffusivity*. If at the point (x, y, z) 
there is a source of heat supplying in time dt a quantity F (x, y, z,t) dxdydzdt 

* This name was suggested by Lord Kelvin. A useful table of the quantities K, c, p and x is 
given in Ingersoll and Zobel's Mathematical Theory of Heat Conduction (Ginn & Co., 1913). 



The Drying of Wood 121 

of heat to the volume element dxdydz, a term F (x, ?/, z, t) must be added 
to the right-hand side of the equation. 

A similar equation occurs in the theory of diffusion ; it is only necessary 
to replace temperature by concentration of the diffusing substance in order 
to obtain the derivation of the equation of diffusion. The quantity of 
diffusing substance conducted from place to place now corresponds to the 
amount of heat that is being conducted. The theory of diffusion of heat was 
developed by Fourier, that of a substance by Fick. In recent times a 
theory of non-Fickian diffusion has been developed in which the coefficient 
K is not a constant. Reference may be made to the work of L. F. Richard- 
son*. 

2-14. An equation similar to the equation of the conduction of heat 
has been used recently by Tuttle f in a theory of the drying of wood. It 
is known that when different parts of a piece of wood are at different 
moisture contents, moisture transfuses from the wetter to the drier regions ; 
Tuttle therefore adopts the fundamental hypothesis that the rate at which 
transfusion takes place transversely with respect to the wood fibres or 
elements is proportional to the slope of the moisture gradient. 

This assumption leads to the equation 

d_e dW 
di W 

where 9 is moisture content expressed as a percentage of the oven-dry 
weight of the wood and It 2 is a constant for the particular wood and may 
be called the transfusivity (across the grain) of the species of wood under 
consideration. 

From actual data on the distribution of moisture in the heartwood of 
a piece of Sitka spruce after five hours' drying at a temperature of 160 F. 
and in air with a relative humidity of 30 %, Tuttle finds by a computation 
that h 2 is about 0-0053, where lengths are measured in inches, time in 
hours and moisture content in percentage of dry weight of wood. 

The actual boundary conditions considered in the computation were 

6 = at x - 0, 9 = at x - 1, = when t - 0. 

A more complete theory of drying has been given recently by E. E. 
LibmanJ in his theory of porous flow. He denotes the mass of fluid per 
unit mass of dry material by v and calls it the moisture density. The 
symbols p, a, r are used to denote the densities of moist material, dry 
material and fluid respectively and ft is used to denote the coefficient of 
compressibility of the moist material. 

* Proc. Roy. Soc. London, vol. ex, p. 709 (1926). 

f F. Tuttle, Journ. of the Franklin Inst. vol. cc, p. 609 ( 1925). 

t E. E. Libman, Phil. Mttg. (7), vol. iv, p. 1285 (1927). 



122 Applications of the Integral Theorems of Gauss and Stokes 

The rate of gain of fluid per unit mass of dry material in the volume V 
is the rate of increase of v, where v is the average value of v in F. If w 
is the mass of dry material in volume V of moist material and f n = mass 
of fluid flowing in unit time across unit area normal to the direction n we 

have 



dv V 
therefore ~-. = --- div/. ...... (B) 

ot w 



Now, the mass of fluid in the volume V is wv and the total mass of 
material in V is wv -j- w and is also pF, hence 



and (B) gives the equation < 

'dv I + v 

3t=~ >" 
for the interior of the porous body. 

If EdS denotes the mass of fluid evaporating in unit time from a small 
area dS of the boundary of the porous body the boundary condition is 

fn = E. 

The flow of fluid in a porous material may be regarded as the sum of 
three separate flows due respectively to capillarity, gravity and a pressure 
gradient caused by shrinkage. We therefore write, for the case in which 
the z axis is vertical and p is the pressure, 

- dv j dz 7 dp 

/.--Kfr-kgrfc-kft, 

where K and k are constants characteristic of the material. 

\-\-v 
Consider now a small element of volume 8w at the point P (#, y, z), 

the associated mass of dry material being 8w and the volume per unit mass 
of dry material 1 + v 



Then "- = " 

dv dv 

dp dp dV dp d /I -j- v\ 
fo~Wdv ss dVdv(~j~)' 

1 dV 

But by definition B = ^ 7 ~ T - , 

K ap 

therefore $=- l d f 1 + ^- ! d /-- X + v 



- -j- _ 
Vfidv\ p 

1 rf A 1 + 



\ 1 d A 1 -f v\ 
) = _ - (w IU . ) 
) fidv\ 6 p /' 



The Heating of a Porous Body 123 

^- d<f> 

Putting dl-X- 

we have ..- 



or / - - f -- f - - kar 

Jx ~ dx' Jv ~ dy' h ~ c!z~ J ' 

div/= - V*<f>, 

and so t p ^ = W, 

1 + t> 3< r 

wliile the boundary condition takes the form 

*- + IS + tgr* = 0. 



It should be mentioned that in the derivation of this equation the 
material has been assumed to be isotropic. 
In the special case of no shrinkage we have 

p = a (1 -f v), ,- = K , cf> = JSTi; 4- const., 
and the equation for v becomes 



which is similar in form to the equation of the conduction of heat. The 

boundary condition is ^ ~ 

, r ov n 7 oz 
A" + E 4- kgr - 0. 
dn on 

2-15. The heating of a porous body by a warm fluid*. A warm fluid 
carrying heat is supposed to flow with constant velocity into a tube which 
contains a porous substance such as a solid body in a finely divided state. 
For convenience we shall call the fluid steam and the porous substance 
iron. The steam is initially at a constant temperature which is higher than 
that of the iron. The problem is to determine the temperatures of the iron 
and steam at a given time and position on the assumption that the specific 
heats of the iron and steam are both constant and that there are no* heat 
exchanges between the wall of the tube and either the iron or steam, no 
heat exchanges between different particles of steam and no heat exchanges 
between different particles of iron. The problem is, of course, idealised by 
these simplifying assumptions. We make the further assumption that the 
velocity of the steam is the same all over the cross-section of the pipe. 
This, too, would not be quite true in actual practice. 

Let U be the temperature of the iron at a place specified by a co-ordinate 
x measured parallel to the axis of the pipe, V the corresponding temperature 

* A. Anzelius, Zeit*. f. ang. Math. u. Mech. Bd. vi, S. 291 (1926). 



124 Applications of the Integral Theorems of Gauss and Stokes 

of the steam. These quantities will be regarded as functions of x and I only. 
This is approximately true if the pipe is of uniform section so that the 
cross-sectional area is a constant quantity A. 

Let us now consider a slice of the pipe bounded by the wall and two 
transverse planes x and x f- dx. At times t and t -f dt the heat contents 
of the iron contained in this slice are respectively 

uUA dx and u ( U + ~ dt] A dx, 

V at J 

where u is the quantity of heat necessary to raise the temperature of unit 
volume of the iron through unit temperature. Thus the quantity of heat 
imparted to the iron in the slice in time dt is 

dQ,= uA~dtdx. 
ct 

Similarly, at time t the heat content of the vapour in the slice is rVA dx 

( dV \ 
and at time / h dt it is v ( V -f -^ - dt] A dx, where v is a quantity analogous 

to u. 

With the steam flowing across the plane x in time dt a quantity of heat 
vVAcdt is brought into the slice where c is the constant velocity of flow. 

In the same time a quantity of heat v[V+ - dx]Acdt leaves the slice across 

V ox / 

the plane x -f <tx. The steam has thus conveyed to the iron a quantity of 

heat ~ ir - Tr 

i^ (3V 5V\ A 7 7 

dQ 2 = -- v -,- -I- c ~ }A dtdx. 

\ ot ox/ 

In accordance with the law of heat transfer that is usually adopted the 
quantity of heat transferred from the steam to the iron in the slice in time 

dils d#3 = k(V- U) A dtdx, 

where k is the heat transfer factor for iron and steam. We thus have the 

equations 



With the notation a = k/cv, b = k/cu and the new variables 

= ax, r = b (ct x), 
the equations become 

W-u-v du -v-u 

3{~ ' ST y 

These equations imply that the quantity A (, r) defined by 

b(S,T) = #**(V-U) 

is a solution of the partial differential equation 






Laplace's Method 125 

The supplementary conditions which will be adopted are 

tf (0)= U 19 7(0, T)= F 1? 

where D^ and F x are constants. The equation (A) then gives 
f/(0, r)= ^-(Fi- 1^)6-', 
F(,0) = C/x+CFj- tfje-f, 

and so the supplementary conditions for the quantity A are 
A (f , 0) = A (0, r) = F! - t/! = IF, say. 

2-16. Solution by the metfwd of Laplace. The equation (A) may be 
solved by a method of successive approximations by writing 

A- AO + A x + A 2 + ..., 

where \ = W and ~ *- = A^^. 

This gives A - JF/ [2 V(fr)J, 

V - U= TFe-<^>/ 



rax 

=. V l - H'c- 6 <*-*> e~ s / [2 V{^ (c^ - x)}] ds, 

Jo 

= U l + TFe~ a ^ f C< ^c-'/o [2 

J o 



u = c/i + 

For x> ct the solution has no physical meaning but for such values of oc 
the iron has not yet been reached by the steam and so U C^. 

As t -> oo we should have U -+ V 19 V -> V 1 ; this condition is easily seen 
to be satisfied, for our formula for V U indicates that V U -> and 
U -> F t because 

[ e~ s 7 [2^/(axs)] ds = e ax . 
Jo 

The properties of the solution might be used, however, to infer the value 

of this integral. 

EXAMPLE 

Prove that if E (r) = 2 -^- 3 , 

the differential equation ^^^ = V 

^ . dxdydz 

fV (z 
is satisfied by V=\ I </>(v, w)E{x(y v)(z w)}dvdw 

[z (x 
-f I i/i(w,u) E {y(z w)(x u)}dwdu 

[x fy 

Jo Jo 

+ I X P (u) E {(x - u) yz} du + ( V Q(v) E {(y - v) zx} dv 
J J 

+ [ Z E (w) E {(z - w) xy} dw + SE (xyz). 
[T. W. Chaundy, Proc. London Math. Soc. (2), vol. xxi, p. 214 (1923).] 



126 Applications of the Integral Theorems of Gauss and Stokes 

2-21. Riemanris method. Let L (u) be used to denote the differential 
expression a 

2 U CU , OU 

~ a 4- #0 + 63 + cu y 

dxdy dx dy 

where a, b, c are continuous (Z), 1) in a region ^? in the (x, y) plane. The 
adjoint expression L (v) is defined by the relation 



where M and N are certain quantities which can be expressed in terms of 
u, v and their first derivatives. Appropriate forms for L, M and N are 

S 2 v d d 

L (v) = - -- ^ (at;) - ~ (6v) + cv, 
dxdy 9^; ty 

nf I / du dv\ 19, v ^ , v 

If = aw + v-u = (uv) - uP (v), say, 



'w 9i; 

2 aa . - 

, T> 7 x 9v ^ 7 . dv ^ , 

where P (v) = - - av, Q (v) = x 60;. 

Now if C' is a closed curve whiclMi^s entirely within the region R and 
if both u and v are continuous (D, 1) in ,&, we have by the two-dimensional 
form of Green's theorem 



(IM + mN) ds^ + dxdy = [vL (u) - ul (v)] dxdy, 

where /, m are the direction cosines of the normal to the curve C and the 
double integral is taken over the area bounded by C, and so will be ex- 
pressed in terms of the values of u and its normal derivative at points of 
the curve F, for when u is known its tangential derivative is known and 

^ - and XT- can be expressed in terms of the normal and tangential de- 

rivatives. If (X Q , y ) are the co-ordinates of the point A the function v 
which enables us to solve the foregoing problem may be written in the 

form , x 

v = g (x, ?/; x , # ), 

and may be called a Green's function of the differential expression L (u). 

This theorem will now be applied in the case where the curve C consists 
of lines XA, A Y parallel to the axes of x and y respectively and a curve I 1 
joining the points Y and X. 

Using letters instead of particular values of the variable of integration 
to denote the end points of each integral, we have when L (u) = 0, L (v) = 0, 



t X Ndx- \ 

J A J 



Riemann's Method 127 

[A [A 

Now M dy = 4 \(uv) Y (uv) A ] + uP (v) dy, 

JY JY 

f A f x 

and Ndx = J [(w)x - (uv) 4 ] - uQ (v) dx, 

JA J A 

rx 
therefore (uv) A = i [(w)v + (wv)r] + (IM +.mN) efe 

J Y 

CA rx 

-f ?/P (v) dy wQ (v) eu-. 
J i' .' .4 

If now the function v can be chosen so that P (v) = on ^4 7 and Q (v) = 
on AX, the value of ^ at the point A will be given by the formula 

rx 

(uv) A = J .[(uv) v + (^')r] -f (J-W + mN) ds\ 

i Y 

It should be noticed that if u is not a solution of L (u) ~ but a solution of 



the corresponding expression for u is 

(uv) A = $ [(uv)x + (uv) Y ] + I (IM -h mN) ds -f- | U/(.r, ? 

r 

An interesting property of the function y may be obtained by consider- 
ing the case when the curve F consists of a line YB parallel to AX and a 
line BX parallel to YA. We then have x 

[ A (IM -f mN) ds = \ X Mdy- {" Ndx, 

J Y J B J Y 



also M = - ^ V (wv) + ^^ M, ^ (w) = a ^ + 

N ^~l L (uv] + "Q (u] - Q (u] = ^x+ 

[ B [ B - 

we have Ndx = \ [(uv) Y (uv) E ] + vQ (u) dx, 

J Y J Y 



J 



Mdy - | [(uv) B - (uv) x ] + f - vP (u) dy. 
B J B 



CB r 

Hence (^^)x = (uv) B vQ (u) dx + 

JY J 



Now let a function u = h (x,y\x l , y^) be supposed to exist such that 
Z/ (w) = 0, P (w) = on JBX, Q (u) = on J?7 ; the co-ordinates a?!, ^ being 
those of B. The formula then gives 

(uv) A - (ttt;)^. 

Choosing the arbitrary constant multipliers which occur in the general 
expressions for g and h, in such a way that 

9 (z > y l x 0t y ) - 1, h (x lf y^ x l9 yj = 1, 



128 Applications of the Integral Theorems of Gauss and Stokes 
the preceding relation can be written in the form 

h (x , y ; x l9 yj = g (x ly y^ X Q , y Q ). 

When considered as a function of (x, y) the Green's function g satisfies 
the adjoint equation L (v) = 0, but when considered as a function of 
(X Q , y Q ) it satisfies the original equation expressed in the variables x Qy y Q . 

EXAMPLE 

Prove that the Green's function for the differential equation 



is 

and obtain Laplace's formula 



tt = f*/ 

Jo 



for a solution which satisfies the conditions 



.5- = </> (x) when x = 0. 
oy 

2-22. Solution of the equation ^ 



Let the curve B consist of a line A 1 A 2 parallel to the axis of x and two 
curves C l7 C 2 starting from A ly A z respectively and running in an upward 
direction from the line A 1 A 2 . Let S denote the realm bounded by the 
portion of B below a line y parallel to the axis of x and the portion of B 
which lies between C l and (7 2 . When y is replaced by the parallel lines 
y , y the corresponding realms will be denoted by S Q and 8' respectively. 
The portions of B below the lines y, y , y will be denoted by /?, /? and ft' 
respectively. The equation of A 1 A 2 will be taken to be y = y lt 

We shall now suppose that y and y' both lie below y and that z is a 

solution of (A) which is regular in S , regularity meaning that z, x-*, ~ - 

OX 01/ 

a 2 - 

and ,5-j are continuous functions of x and y in the realm S . 

OX 

The differential expression adjoint to L (z) is L (f), where 



and we have the identity 

t [L (,) -/(*, )] - Z L ( = - 



Generalised Equation of Conduction 129 

Hence if. L (z) = f (x, y), 

and L (t) = 0, 

we have ^ | [ | - **] - | [to] - /- 0. 

Let us now write = 2 (, 77),. r = (, T;) and integrate the last equation 
over the region S', then 

/, - J r [* + Mr '*)*] -//, 

In this equation we write 

r ( ,) = T [*, y; f, 77] EE (y - 77)-* e^p [- (* - g)*/(y - 77)], 
and we take y to be a line which lies just below the line y which passes 
through the point (x, y). Our aim now is to find the limiting value of the 
integral on the left as y -> y. 

By means of the substitution % = x -+- 2u\/(y 77) this integral is 
transformed into 

/tta 

2 s [a: -f 2tt (y - 17)*, 77] e- wt ^. 

.' MI 

If the equations of the curves O l , <7 2 are respectively 

ar = c x (y), a; - c 2 (y), 
the limits of the integral are respectively 

^i = [Ci (17) - a;]/2 V(y ~ T?), ^2 = [Cz (>?) - x]/2 ^/(y - y). 
If the point (x, y) lies within S we have 

u t -> oo, i^ 2 -* + oo as y x -> y and 17 -+ y ; 
if it lies outside $ we have 

u -> w 2 -> oo as y x -v y and ?} -+ y. 

Finally, if the point (x, y) lies on either 'C 1 or (7 2 one limit is zero, thus 
we may have either u t -> 0, u 2 -+ oo , or u -> oo, w 2 -^ 0. 

The Umiting value obtained by putting rj ~ yin the integral is 2z (x, y)\/7r 
in the first case, zero in the second case and z (x, y) \Ar in the third. Hence 
when the limiting value is actually attained we have the formula 

z (x y y) [2V", or 



VI = f \Tdf + (T - ^) ^1 - f f 

^ L \ ^? c/f / J .-Js 



The transition to the limit has been carefully examined by Levi*, 
Goursatf and Gevreyf. The last named has imposed further conditions 

* E. E. Levi, AnnaLi di Matematica (3), vol. xiv, p. 187 (1908). 

t E. Goursat, TraiU # Analyse, t. in, p. 310. 

t M. Gevrey, Jvurn. de Mathdmatiques (6), t. ix, p. 305 (1913). See also Wera Lebedeff, Diss. 
Oottingen (1906); E. Holmgren, Arkiv for Matematik, Astronomi och Fyaik, Bd. in (1907), Bd. iv 
(1908); G. C. Evans, Amer. Journ. Math. vol. xxxvn, p. 431 (1915). 



130 Applications of the Integral Theorems of Gauss and Stokes 

in order to establish the formula in the case when (x, y) lies on either C l 
or C 2 . His conditions are that, if (f> (s) is continuous, 

lim [c, (y) - c, (rj)] (y - ^ = (p = 1, 2), 

r)->y 

that 

f [c, (y) - c, (*)] (y - *)-* <f> (s) ds . exp [- {c p (y) - c p (s)}*/4 (y - s)} 

-' >) 

should exist and that the functions q (y), c 2 (y) be of bounded variation. 
It may be remarked that the line integrals in this formula are particular 

solutions of the equation ~ = ~ 2 , while the integral 



_ I I ? (x, y; 

/ 7T .' ' 



, _ 

^ \/ 7T .' ' >o 

is a particular solution of the equation (A). Sufficient conditions that this 
may be true have been given by Levi and less restrictive conditions have 
been formulated by Gevrey. The properties of the integrals 

f f dT 

I(x,y) = T(x ) y^ ) r ] ) ( f>(7 ] )dr )) J(x,y) = ~ <f> (77) dy 
. o c ^"^ 

have also been studied, where O is a curve running from a point on the 
line y = y l to a point on the line y = y. It appears that when the point P 
crosses the curve C at a point P the integral / suffers a discontinuity 
indicated by the formula 

lim (J P - J P ) - <f> Po VTT, 
P->P O 

the sign being + or according as P approaches P from the right or the 
left of the curve C. 

In this formula </> denotes any continuous function and a suffix P is 
used to denote the value of a function of position at the point 'P. 

A Green's function for the region 8 may be defined by the formula 

G (x, y; ,-n) = T (x, y; f 9 r))-H (x, y; 77), 

r)%Pf T^T-f 

where H (x, y; , ??), which satisfies the equation -^ +-* = 0, is zero on 

c or) 
y when considered as a function of f and 77, which is regular and which 

takes the same values as T (x, y ; , T?) on the curves C t and C 2 . The function 

n * *u . x- 9 2 # W 3*G , SG A . 

G satisfies the two equations ^ = - , -x> 2 - + ^ = 0, is zero when x = c l (y) y 

when ^ = Cj (77), when x = c 2 (y) and when f = c 2 (77), and is positive in $. 
With the aid of this function a formula 



-ff 

J J S 



The Green's Function 131 

may be given for a solution of (A) which takes assigned values on /?. The 
problem of determining O is reduced by Gevrey to the solution of some 
integral equations. 

Fundamental solutions of the equations 

dz _ a 3 z d*z _ d*z 
dy dx 3 ' dy* dx* 

have been obtained by H. Block* and have been used by E. Del Vecchiof 
to obtain solutions of the equations 

dz d*z_ f d*z d*z _ f 

dy ~dx*~ J (X ' y) ' dy* ^8x*~ J (X > y} ' 

EXAMPLES 
1. Show by means of the substitutions 



that the integral 

' ' (x - f)*(y - ?)-exp[- (x - 



II. 



has a meaning when p + 1 > and p 2q + 3 > 0, / being an integrable function . 

[E. E. Levi.] 

2. Show that by means of a transformation of variables 

x' = x' (x, y), y' = y 

the parabolic eq nation ~ -f a ~ -f 6 - 4- cz -f / = 

^ n dx 2 dx dy 

may be reduced to the canonical form 

a 2 2 dz dz 

a r /2 a , = a ^~, + cz +f. 
dx 2 dy dx 

Show also that the term p , may be removed by making a substitution of type z uv 
ox 

and that the term involving u will disappear at the same time if 

d' 2 a da da rt dc 
a - /2 = a a , 5- , -f 2 . , . 
dx 2 dx dy dx 

3. If a, b and c are continuous functions in a region R a solution of 

(6<0) 



dx 2 dx dy 

which is regular in R can have neither a positive maximum nor a negative minimum. Hence 
show that there is only one solution of the equation which is regular in R and has assigned 
values at points of a closed curve C lying entirely within R. 

[M. Gevrey.] 

2-23. Green's theorem for a general linear differential equation of the 
second order. Let the independent variables x lf x 2) ... x m be regarded as 
rectangular co-ordinates in a space of ra dimensions. The derivatives of 

* Arkivf. Mat., Ast. och Fysik, vol. vn (1912), vol. vm (1913), vol. ix (1913). 
t Mem. d. R. Accad. d. Sc. di Torino (2), vol. LXVI (1916). 

9-2 



132 Applications of the Integral Theorems of Gauss and Stokes 
a function u with respect to the co-ordinates may be indicated by suffixes 

^\ ^2 

written outside a bracket, thus (u) 2 stands for x and (u) 23 for 5^ . 

(7it*2 @&2 ^*^3 

We now consider the differential equation 

L (u) = S S ^ rs Mr, + S r (tt)r -f Fw = 0, 

>=ls-l r-1 

/I _ J 

^rs -^sr* 

where the coefficients A rs , B r , F are functions of x l9 x 2 , ... x m . 
The expression Z (v) adjoint to L (u) is 

L(v)= S 2 (4 rs v) rf - S(fl r t;) r + J T t;, 

r-l sl r-1 

and we have the identity 

vL(u)-uL(v)= S (^ r ) r , 

r-l 

m 77t r m 

where Qr = - u 2 ^4 rs (v\ s + v S -4 rs (u), 4- ^v r S (,4 rs 



The ^-dimensional form of the theorem for transforming a surface 
integral into a volume integral may be written in the form 

rr m rr m 

\ \ V(Q r ) r dx, ... dx m =-\\ Zn r Q r dS, 

where n l9 n 2 , ... n m are the direction cosines of the normal to the hyper- 
surface S, the normal being drawn into the region of integration. Hence 
we have the equation 

f f r r 

[vL (u) - uL (v)] dx l ... dx n = \ \{v D n u - uD n v uvP n ] dS 9 

vi m 

where D n (u) = 2 n s A rs (u) r , 

m r m n 

in 

Let us write S n s A rs = A^ r , 

where v l9 v 2 , ... v m are the direction cosines of a line which may be called 
the conormal*, then 

D n (u) A 2 v r (u) r = A (u) v . 

r-l 

2-24. The characteristics of a partial differential equation of the second 
order. Let the values of the first derivatives (u) l9 (u) 2 , ... (u) m be given at 
points of the hypersurface 6 (x l9 x 2 , ... x m ) = 0. If dx l9 dx 2 , ... dx m are 
increments connected by the equation 

(9)idx l + (0) 2 dx 2 4- ... (0) m rfa? w = 0, 
* This is a term introduced by R. d'Adh^mar. 



Characteristics 133 

and if (0^ ^ 0, we may regard the increments dx 2 , dx 3 , ... dx m as arbitrary, 

and since , r/ x n , v , , v , , . 7 

= (w^a-rj -f (u) p2 dx 2 -f ... (M)p m &r m , 



the quantities (0)! (M)^ (6) s (u) pl 

may be regarded as known. Similarly the quantities 

(*)i WIP - ()p (")n 
may be regarded as known and so the quantities 



may be regarded as known. Substituting the values of (u) ps in the partial 
differential equation 

m m in 

L(u)= S 2 ^4 rs (w)r 4- 2 B r (u) r -f /V = 0, (I) 

r-l s-1 r-1 

we see that we have a linear equation to determine (u) n in which the 
coefficient of (u) n is 

A = S S 4 r ,(0) r (0),. (II) 

If this quantity is different from zero the equation determines (u) n 
uniquely, but if the quantity A is zero the equation fails to determine 
(u) n and the derivatives (u) are likewise not determined. In this case the 
hypersurface 6 (x 11 x 2 , ... x m ) = is called a characteristic and the dif- 
ferential equation A = is called the partial differential equation of the 
characteristics. 

The equations of Cauchy's characteristics for this partial differential 
equation of the first order are 

dxi __ dx 2 _ dx m 

and these are called the bicharacteristics of the original partial differential 
equation. All the bicharacteristics passing through a point (a^ , a: 2 , ... x m ) 
generate a hypersurface or conoid with a singular point at (xf, x 2 , ... # m ). 
When all the quantities A^ are constants this conoid is identical with the 
characteristic cone which is tangent to all the characteristic hypersurfaces 
through the point (x-f*, x 2 , ... x m ). 

For the theory of characteristics of equations of higher order reference 
may be made to papers by Levi* and Sanniaf. These authors have also 
considered multiple characteristics and Sannia gives a complete classifica- 
tion of linear partial differential equations in two variables of orders up to 5. 

* E. E. Levi, Ann. di Mat. (3 a), vol. xvi, p. 161 (1909). 

t G. Sannia, Mem. d. R. Ace. di Torino (2), vol. LXIV (1914); vol. LXVI (1916) 



134 Applications of the Integral Theorems of Gauss and Stokes 

2-25. The classification of partial differential equations of the second 
order. A partial differential equation with real coefficients is said to be of 
elliptic type when the quadratic form 



is always positive except when X 1 = X 2 = ... == X n = 0. 

The use of the words elliptic, hyperbolic and parabolic seems natural in 
the case n = 2, the term to be used depending upon the nature of the conic 
A n X^ + 2A 12 X 1 X 2 + A 22 X 2 * = 1. 

For a non-linear equation F (r, ,9, t, p, q, z) = 

r _ a s z 3*z_ _ 3 2 z _dz 

I """ a* 2 ' ' "" dxdy' ~ dy*' P ~~ dx' 
there is a similar classification depending on the nature of the quadratic 

form 

dF dF dF 

\ * _J_ 9 V X 4- Y 2 Vr 

Al dr +^1^29^-+ A s - dt . 

When n > 2 the classification is not so simple ; for instance, when n 3 
it may be based on the different types of quadric surface and it is known 
that there are two different types of hyperboloid. 

The word ellipsoidal might be used in this case instead of elliptic, but 
it seems better to use the same term for all values of n because the im- 
portant question from the standpoint of the theory of partial differential 
equations is whether the equation is or is not of elliptic type. For an 
equation of elliptic type the characteristics are all imaginary and this fact 
has a marked influence on the properties of the solutions of the equation. 
When n = 2 typical equations of the three types are 
d 2 u d*u du , du 



d*u du , du ^ n i i. x 

~~- + a ~ 4- * -f cu (hyperbolic), 
dxdy dx dy J ; 

d 2 u du . du . . , T , 

- 2 -f a - -h o ~ + cu = (parabolic). 

A notable difference between elliptic and hyperbolic equations arises 
when a solution is required to assume prescribed values at points of a 
closed curve and be regular within the curve. For illustration let us con- 
sider the case when the curve is the circle x 2 -f y 2 = 1. If the boundary 
condition is V sin 2nO when x cos 0, y = sin 6, where n is a positive 

? 2 K 
integer, there is no solution of the equation ~ ~ which is continuous 

(D, 1) and single- valued within and on the circle*, but there is a regular 

* When . ! there is a solution V = 2y (1 y 2 )^ which satisfies the boundary condition but 

dV 

its derivative - is infinite on the circle. 
cy 



Classification of Equations 135 

927 327 
solution of -=- 2 + O-T = > namely, V = r 2n sin 2n0. On the other hand, 

if the boundary condition is V = sin (2n + 1) 9, there is a solution of 

3 2 F * 

-----= of type V = f(y) which satisfies the conditions and is single- 

valued and continuous in the circle, but this solution is not unique because 



V = 1 x 2 y 2 is a solution of ~ ~- = which is zero on the circle and 
y dxdy 

single -valued and continuous inside the circle. 

When the solution of a problem is not unique or when there is some 
uncertainty regarding the existence of a solution the problem may be 
regarded as not having been formulated correctly. An important property 
of the boundary problems of mathematical physics is that the correct 
formulation of the problem is indicated by the physical requirements in 
nearly every case. 

2-26. A property of equations of elliptic type. Picard*, Bernsteinj 
and Lich tens tern t have shown that the solutions of certain general 
differential equations of elliptic type cannot have maximum or minimum 
values in the interior of a region within which they are regular. This 
property, which has been known for a long time for the case of Laplace's 
equation, has been proved recently in the following elementary way. 

Let L (u) = T A^ (u)^ 4- S B v (u) v 

1,1 i 

be a partial differential equation of the second order whose coefficients 
AH V) B v are continuous functions of the co-ordinates (x l> x 2 , ... x n ) of a 
point P of an n-dimensional region T. For convenience we shall sometimes 
use a symbol such as u (P) to denote a quantity which depends on the co- 
ordinates of the point P. We can then state the following theorem : 

// u (P) is continuous (), 2) and satisfies the inequality L (u) > every- 
where in T, an inequality of type u (P) < u (P ) can only be satisfied through- 
out T, where P is a fixed internal point, when the inequality reduces to the 
equality u (P) u (P ). Similarly, if L (u) < throughout T, the inequality 
u (P) > u (P ) in T implies that u (P) = u (P Q ). 

The proof will be given for the case n = 2 so that we can use the familiar 
terminology of plane geometry, but the method is perfectly general. 

Let us suppose that L (u) > in T and that u (P ) = M , while u (P) < M 
if P is in T. 

If-u^M there will be a circle C within T such that at some point P 
of its boundary, say at P l9 we have u (P^ = M, whilst in the interior of 
the circle u < M . 

* E. Picard, T mitt tf Analyse, t. ir, 2nd ed., p. 29 (Paris, 1905). 

t S. Bernstein, Math. Ann. Bd. ux, S. 69 (1904). 

J L. Lichtenstein, Palermo Rend. t. xxxm, p. 211 (1912); Math. Zeitschr. vol. xx, p. 205 (1924). 

E. Hopf, Berlin. Sitzungsber. S. 147 (1927). 



136 Applications of the Integral Theorems of Gauss and Stokes 

Let K be a circular realm of radius R whose circular boundary touches 
C internally at P, then with the exception of the point P l , we have every- 
where in K the inequality u < M. 

Next let a circle K 1 of radius /^ < R be drawn so as to lie entirely 
within T. The boundary of K then consists of an arc S t (the end points 
included) which belongs to K and an arc $ which does not belong to K. 
On S t we have the inequality u < M e, where e is a suitable small 
quantity, while on S we have u ^ M. ...... (A) 

We now 'choose the centre of K as origin and consider the function 

h(P) == e-' 1 ' 2 - e- i; -", 
where r 2 = a: 2 -f ?/ 2 and a > 0. If x t = x, x 2 = y and 

(M) - .4w^ -f 2?^ y -f CM,, -f Du x -f J^, 
a simple calculation gives 

e^L (h) = 4a 2 (Ax 2 -f 2&oy -f Cy 2 ) - 2a (A + C -f Dz -f %). 

Since the equation is of elliptic type we have in the interior and on the 

boundary of K . rt ~ 9 ^ 7 ^ 

J J.r 2 -f 2Bxy -f <7?/ 2 > & > 0, 



where k is a suitable constant. By choosing a sufficiently large value of a 

we can make r , 7 , . 

L (//) ^> 

in K l and so L (u 4- 8A) > 

if 8 > 0. We have, moreover, h (P) < when P is on $ , & (Pj) = 0. 

..... (B) 

We now put v (P) = u (P) -f- 8 . h (P), 8 > 0, where 8 is also chosen so 
small that, in view of (A), we have v < M on S z . On account of (A) and 
(B) we have further v < M on S Q . Hence v< M on the whole of the 
boundary of K 1 and at the centre we have v = M . Thus v should have a 
maximum value at some point in the interior of K l . This, however, may 
be shown to be incompatible with the inequality L (v) > 0, for at a place 
where v is a maximum we have by the usual rule of the differential calculus 





for arbitrary real values of A and ^. Now, by hypothesis, 



therefore by the theorem of Paraf and Fejer ( 1-35) 



Av xx + 2Bv xy + Bv yv < 0. 



But the expression on the left-hand side is precisely L (v) since v x == v v 0, 
and so we have L (v) < which is incompatible with L (v) > 0. 

The case in which L (u) < 0, u (P) > u (P ) can be treated in a similar 
way. 



Maxima and Minima of Solutions 137 

In particular, if L (u) = in T, where u is not constant, neither of the 
inequalities u (P) < u (P ), u (P) > u (P ) can hold throughout T when P 
i& an internal point. This means that u (P) cannot have a maximum or 
minimum value in the interior of a region T within which it is regular. 

This theorem has been extended by Hopf to the case in which the 
fr notions A, B, C, D, E are not continuous throughout T but are bounded 
functions such that an inequality of type 

AX 2 + 2BXfM + C> 2 > N (A 2 + n*) > 

holds, with a suitable value of the constant N, for all real values of A and 
H and for all points P in T. 

The work of Picard has also been generalised by Moutard* and Fejerf. 
The latter gives the theorem the following form : 

Let 

n, ... n n 

S a ik (x 1) x 2) ...x n )(u) tk + I>b r (x l9 x 29 ... x n ) (u) r + c (x l9 ...x n )u= 0, 
i,...i i 

a tk ( X l> %2> X n) a Jft ( X \> ^2? %n) 

be a homogeneous linear partial differential equation of the second order 
with real independent variables x l9 x 2 , ... x n and a real unknown function 
u (x l9 x 2) ... x n ). The coefficients 

^ik \%lj *^2> * ^n/9 r \%l j *^2 > ^n/9 ^ (^1 > ^2 > *^n) 

are all real functions which can be expanded in convergent power series of 

yP6S c (x l9 * 2 , ... x n ) - c + Cj^ + ... c n x n + c n x^ -f ... , 

6 r (a?!, # 2 , ... x n ) = 6 r -f & rl #i + ... ft rn ^ n -f b rll xf + .... 

fjfc (^1, ^ - *n) = tA: + GW^l + ..-, 

for \Xi\<*i, I ^2 I < ^ 2 I %n | < , 

where z l9 z 2 , ... n are suitable constants. Then, if 

n, n 

2 a tk y t y k > 
1,1 

for all real values of y l9 y 2 , ... y ny that is, if the quadratic form is non- 
negative, and if c < 0, the differential equation has no solution which is 
regular at the origin and has there either a negative minimum or a positive 
maximum. If, however, the quadratic form is not negative, that is, if 

71, 71 

2 a t *iM* < 

for some set of values rj l9 ^ 2 , ... rj k9 there is always a solution regular at 
the origin which, if c < 0, has either a negative minimum or a positive 
maximum. Thus when c< the requirement that the quadratic form 

* Th. Moutard, Journ. de Vtfcole Poll/technique, t. LXIV, p. 55 (1894); see also A. Paraf, Annates 
de Toulouse, t. vi, H, p. 1 (1892). 
t Loc. cit. 



138 Applications of the Integral Theorems of Gauss and Stokes 

should not be of the non-negative type is a necessary and sufficient con- 
dition for the existence of a negative minimum or positive maximum at 
the origin for some regular solution of the differential equation. 

2-31. Green's theorem for Laplace's equation. Let us now write in 
equation (A) of 2-11 



the equation then takes the form 



dU dV dU dV dU dV\ 



rr dv \jv trrwrj t ( d \ 

U v~ }dS = UV 2 Vdr -Mis- -x-- + ~- 3- 4- -5- -~- 
on/ J J \ox dx dy dy dz dz J 

dV 
where the symbol ~-- is used for the normal component of VF, 

dV dV dV dV 



Interchanging U and V we have likewise 

((v du \jv (VWTJ _L [fiUdVdUdV dUdV 

I K 5 ) a = y\ 2 Udr + - -=-- + - - -^- - - -^- - 

J \ onj ) ) \ox dx dy dy dz dz 

Subtracting we obtain Green's theorem, 



In this equation the functions U and V are supposed to be continuous 
(D, 2) within the region over which the volume integrals are taken. This 
supposition is really too restrictive but it will be replaced later by one 
which is not quite so restrictive. If the functions U and V are solutions 
of the same differential equation and one which has the same characteristic 
as Laplace's equation, an interesting result is obtained. In particular, if 

k*U - 0, 



where k is either a constant or a function of x, y and z, the volume integral 
vanishes and we have the relation 



In the special case when the surface S is a sphere and 

U = f m (r)T m (** g \ V = f n (r)Y n (H Z }, 



where Y m and Y n are functions which are continuous over the sphere and 
fm (r), f n (r) are f unctions such that f m (r) f n ' (r) ^ f m ' (r) f n (r) on the sphere 
we obtain the important integral relation 



which implies that the functions Y m form an orthogonal system. 



Green's Theorem for Laplace's Equation 139 

An appropriate set of such functions will be constructed in 6-34. 
When k is a constant the equations (B) may be derived from the wave- 
equations D 2 ^ = 0, D 2 ^ = by supposing that u and v have the forms 

u = U sin (kct -ha), v = V sin (kct -f /?) 

respectively. When k is real these wave -functions are periodic. When 
k = 0, U and V are solutions of Laplace's equation. 

The equation may also be derived from the equation of the conduction 
of heat, 



by supposing that this possesses a solution of type v = e~* m V(x, y, z). 

Green's theorem is particularly useful for proofs of the uniqueness of 
the solution of a boundary problem for one of our differential equations. 
Suppose, for instance, that we wish to find a solution of Laplace's equation 
which is continuous (Z), 2) within the region bounded by the surface S and 
which takes an assigned value F (x, y, z) on the boundary of S. If there are 
two such solutions U and F the difference W = U V will be a solution 
of Laplace's equation which is zero on the boundary and continuous (D, 2) 
within the region bounded by 8. Green's theorem now gives 

n fw zw , a (T/swY /3WV fiw\*\ i 
== \w --- dS = ( -3 - - ) + K- ) + hr ) dr > 
J dn }[\dx J \dyj \ dz ) J 

and this equation implies that 



dW_ 
y" ' dz ~ Uj 

W is consequently constant and therefore equal to its boundary value zero. 
Hence U = V an|J the solution of the problem is unique. A similar con- 

dW dW 

elusion may be drawn if the boundary condition is ^~ =0 or -~ h h W = 0, 

where h is positive. If the equation is V 2 F -h AF = instead of Laplace's 
equation the foregoing argument still holds when A is negative, for we have 

the additional term A I W 2 dr on the right-hand side. The argument 

breaks down, however, when A is positive. 

In the case of the equation of heat conduction (C) there are some 

similar theorems relating to the uniqueness of solutions. If possible, let 

9?^ _ 
there be two independent solutions v ly v 2 of the equation ^ = *:V 2 t> and 

the supplementary conditions 

v=f(x, y, z) for t = for points within S, 

v ~ <f> (x, y, z, t) on S (t > 0), 

v continuous (J5, 2) within region bounded by S. 



140 Applications of the Integral Theorems of Gauss and Stokes 
Let V = v 1 - v 2 , then F = f or t = within S, V = on S. 

Putting 27 - [ F 2 dr, 

we have ^ = f F 3 /dr = /c I V (V 2 F) dr 

c# J 9 ' 



-I 



T/^F 70 [[fiV\* /3F\ 2 /3 7 

F .- AS - * U ) + br ) + ^ \ dr ' 
dn ] [\Sx/ \dyj \ dz 



Since F = on S the first integral vanishes, and so 

dl /T/^V /3F\ 2 / S V\ 2 ] 7 /,^AX 

~. - - AC ---- ) -I U, - ) + ~ rfr * > 0). 

c# J [_\ do; / \G7// \ c/2 / J 

But 7 = for t = 0, therefore 7 < 0, but on the other hand the integral 
for 7 indicates that 7 > 0, consequently we must have 7=0, F = 0. 

These theorems prove the uniqueness of solutions of certain boundary 
problems but they do not show that such solutions exist. Many existence 
theorems have been established by the methods of advanced analysis and 
the literature on this subject is now very extensive. 

2-32. Green's functions. The solution of a problem in which a solution 
of Laplace's equation or a periodic wave-function is to be determined from 
a knowledge of its behaviour at certain boundaries can be made to depend 
on that of another problem the determination of the appropriate Green's 
function*. 

Let Q(x, y, z- 9 x l9 y l ,z 1 ) be a solution of V 2 G + k 2 G = with the 
following properties: It is finite and continuous (D y 2) with respect to 
either x, y, z or x l , y l , z 1 in a region bounded by a surface S, except in the 
neighbourhood of the point (x 19 y l9 zj where it is infinite like T?" 1 cos kR 
as 7? -> 0, R being the distance between the points (x, y,&) and (x l9 y l ^ z^). 
At the surface S 9 some boundary condition such as (1) G 0, (2) dG/dn = 0, 
or (3) dG/dn -f- hG = is satisfied, h being a positive constant. 

Adopting the notation of Plemeljt and KneserJ, we shall denote the 
values of a function <j> (f, 77, ) at the points (x, y, z) 9 (x l9 y ly z t ) respectively 
by the symbols </> (0) and <(!). 

When a function like the Green's function depends upon the co- 
ordinates of both points it will be denoted by a symbol such as G (0, 1). 
The importance of the Green's function depends chiefly upon the following 
theorem : 

Let U be a solution of 

V 2 C7 + k*U + 47T/ (x, y, z) - 0, (A) 

* G. Green, Math. Papers, p. 31. 

t Monatshefte far Math. u. Phys. Bd. xv, S. 337 (1904). 

J A. Kneser, Die Integralgleichungen und ihre Anwendungen in der mathematischen Physik 
(Vieweg, Brunswick, 1911). 



Green's Functions 141 

which is finite and continuous (Z>, 2) throughout the interior of a region 
bounded by a surface 8 and let / (x, y, z) be a function which is finite 
and continuous throughout !3). We shall also allow / (x, y, z) to be finite 
and continuous throughout parts of the region and zero elsewhere. 

Applying Green's theorem to the region between a small sphere whose 
centre is at (x , y l , zj and the surface S, we have 



Now V 2 (? k 2 G and the first integral on the right may be found by 
a simple extension of the analysis already used in a similar case when 
G = 1/7?, consequently we have the equation 

(1) = (O, I)/ (0)dT + --G d dS ....... (B) 



If U satisfies the same boundary conditions as G on the surface 8 the 
surface integral vanishes and we have* 



'0. (C) 

If, on the other hand, / (x, y, z) = and G = on 8 we have 

(D) 

the value of U is thus determined completely and uniquely by its boundary 

o/nr 

values. Similarly, if the boundary condition is ~ = on S we have 

on 

?, (E) 



and the value of U is determined by the boundary values of dU/dn. 

Finally, if the boundary condition satisfied by is - + hG = 0, we 

have 

(F) 



and U is expressed in terms of the boundary values of ^- f- hU. 

If g (x 2 , 2/ 2 , z 2 ; x, y, z) is the Green's function for the same boundary 
condition as G (0, 1) but for the value a of &, we must also surround the 

* It has not been proved that whenever the function / is finite and continuous throughout D 
the formula (C) gives a solution of (A). Petrmi has shown in fact that when/ is merely continuous 
the second derivatives of the integral may not exist or may not be finite. Ada Math. t. xxxr, p. 127 
(1908). It should be remarked that Gauss in 1840 derived Poisson's equation (2-61) on the 
supposition that the density function/ is continuous (/>, 1). With this supposition (C) does give 
a solution of (A). Poisson's equation and the solution of (A) are usually derived now for the case of a 
function / which satisfies a Holder condition. See Kcllogg's Foundations of Potential Theory, 
ch. vi. 



142 Applications of the Integral Theorems of Gauss and Stokes 

point (# 2 , y*-, z*) by a small sphere when we apply Green's theorem with 
U (x, y, z) = g (2, 0). We then obtain the equation 

/2 _ (jl^ ,' 

g (2, 1) = G (2, 1) - L_J g (2 , 0) G (0, 1) dr, ....... (G) 



This may be regarded as an integral equation for the determination of 
g (2, 0) when G (0, 1) is known or for the determination of G (0, 1) when 
g (2, 1) is known. In some cases the Green's function for Laplace's equation 
(k = 0) can be found and then the integral equation can be used to calculate 
g (2, 0) or to establish its existence. The Green's function for Laplace's 
equation, when it exists, is unique, for if G (0, 1), H (0, 1) were two different 
Green's functions the function 

V (0) = G (0, 1) - H (0, 1) 

would be continuous (D, 2) throughout the region bounded by the surface 
S and satisfy the boundary condition that was assigned, but such a function 
is known to be zero. 

For small values of a 2 the function g (2, 0) can be obtained by expanding 
it in the form g ( ^ Q) = fl (2> Q) + ^ ( ^ Q) + .......... (R) 

The first term is the corresponding Green's function for Laplace's 
equation and is known, the other terms may be obtained successively by 
substituting the series in the integral equation (with k = 0) and equating 
coefficients of the different powers of a 2 . 80 long as the series converges 
this method gives a unique value of g (2, 0). The value of g (2, 0), if it 
exists, will certainly not be unique when a 2 has a singular or characteristic 
value for which the "homogeneous integral equation" has a solution </> 
which is different from zero. In this case 

4rr</> (1) = (a 2 - i) JV (0) G (0, 1) rfr , ...... (I) 

and the formula (C) indicates that this function < (0) = U (x, y, z) is a 
solution of V 2 /-fcr 2 t/=0, which satisfies the assigned boundary con- 
ditions and the other conditions imposed on U. The solutions of this type 
are of great importance in many branches of mathematical physics, 
particularly in the theory of vibrations, and have been discussed by many 
writers. 

The characteristic values of a 2 are called Eigenwerte by the Germans 
and the corresponding functions </> Eigenfunktionen. These terms are 
now being used by American writers, but it seems worth while to shorten 
them and use eit in place of Eigenwert and eif in place of Eigen- 
funktion. The same terms may be used also in connection with the 
homogeneous integral equation (I). In discussing this equation it is 
convenient, however, to put k = 0, so that G becomes the Green's function 



Symmetry of a Green's Function 143 

for Laplace's equation and the assigned boundary conditions. Denoting 
this function by the symbol 4-rrK (0, 1) we have the integral equation 

(0) K (0, 1) dr 

for the determination of the solution of V 2 <f> + cr 2 <f> = and the assigned 
boundary conditions, that is, for the determination of the eifs and eits. 
The function K (0, 1) is called the kernel of the integral equation; it has 
the important property of symmetry expressed by the equation 

K (0, 1) = K (1, 0). 

This may be seen as follows. 

If we put 4rr/ (0) = (o- 2 k 2 ) g (0, 2) in the formula (B) and proceed 
as before, Green's theorem gives 

g (I, 2) = (2, 1) - -^^ /</ (0, 2) O (0, 1) dr . 

Putting o- = k and comparing this equation with the previous one we 
obtain the desired relation. When k ^ the relation gives 

0(1, 2) = 0(2,1). 
When the boundary condition is ~- = this result is equivalent to one 

(j'Yl 

given by Helmholtz in the theory of sound. If \fj (0) is an eif corresponding 
to an eit v 2 which is different from a 2 we have 



I (1) = v 2 (0) K (0, 1) dr , 
and, if the order of integration can be changed, 

K(0, 



= < (0) I (0) dr. = v (1) * (1) drj. 
Hence the eifs < and /r satisfy the orthogonal relation 



This result may be used to prove that the eits a 2 are all real. If, 
indeed, a 2 were a complex quantity a + if! the corresponding eif <f> (0) 
would also be a complex quantity x (0) + i<*> (0), and since K is real the 
function (0) = x (0) *<*> (0) would be an eif corresponding to the eit 
v 2 = a i/J, and the orthogonal relation 



= jt (0) (0) <*T O = |{[ X (O)] 2 + [co (0)]*} dr. 
would be satisfied. This, however, is impossible because the integrand is 



144 Applications of the Integral Theorems of Gauss and Stokes 
either zero for all values of the variables or positive for some, but it is zero 
only when x (0) - , (0) - 0, * (0) - 0. 

To prove that the eits are all positive we make use of the equation 



* j U'dr = - f U Vffrfr- f \(f-) 2 4- (^) 2 + (f )'] dr. 
J J J l\ox/ \dy / \dz / J 

The Green's function is usually found in practice by finding the eifs 
and eits directly from the differential equation and then writing down a 
suitable expansion for G in terms of these eifs. The question of convergence 
is, however, a difficult one which needs careful study. The method has been 
used with considerable success by Heine in his Kugelfunktionen, by Hilbert 
and his co-workers, by Sommerfeld, Kneser and Macdonald, 

2-33. Partial difference equations. The partial difference equations 
analogous to the partial differential equations satisfied by conjugate 
functions are 7/ _ ., ,,, _ -, 

u x "Hi My ~ v x> 

and these lead to the equations of 1-62 

u xx -f- u vv = 0, V x z 4- Vyy = 0, 

which are analogous to Laplace's equation. These difference equations 
have been used in recent years to find approximate solutions of Laplace's 
equation when certain boundary conditions are prescribed* and also to 
establish the existence of a solution corresponding to prescribed boundary 
conditions. 

Let us consider, for instance, a square whose sides are x 2 h, 
y -{_ 2h and let us introduce the abbreviations 

a = h, b = 2h, a = h, j8 = 2h, u (x, y) = (xy), 
u(b,y)=(y), u(p,y) = (y); u(x,b)=[x], u (x, p) = [x], 
then we have eight non-homogeneous equations (fi-A-equations) 

- (Oa) - (0) + 4 (aa) = (a) -f [a], (Oa) + (aO) - 4 (aa) - (a) -f [a], 

- (0) - (Oa) -f- 4 (aa) - (a) -f [a] , (Oa) -f (aO) - 4 (aa) - (a) + [a], 
(aa) + (00) -f (aa) - 4 (aO) = - (0), (act) + (00) + (aa) - 4 (aO) - - (0), 
(aa) + (00) + (aa) - 4 (Oa) = - [0], (aa) + (00) -f (aa) - 4 (Oa) = - [0], 

and one homogeneous equation (A-equation) 

(Oa) 4- (aO) 4- (Oa) 4- (aO) - 4 (00). 

The first step in the solution is to eliminate the quantities (aa), (aa), 
(aa), (aa) which do not occur in the ^-equation. This gives the equations 
4 (00) 4- (Oa) 4- (Oa) - 14 (aO) 4- (a) 4- (a) 4- [a] 4- [a] 4- 4 (6) - 0, 
4 (00) 4- (Oa) 4- (Oa) - 14 (aO) 4- (a) 4- (a) 4- [a] 4- [a] 4- 4 (0) = 0, 
4 (00) 4- (aO) 4- (aO) - 14 (Oa) 4- (a) 4- [a] 4- (a) 4- [a] + 4 [0] = 0, 
4 (00) 4- (aO) 4- (aO) - 14 (Oa) 4- (a) 4- [a] 4- (a) + [a] + 4 [0] = 0. , 

* L. F. Richardson, Phil. Trans. A, vol. ccx, p. 307 (1911); Math. Gazette (July, 1925). 



Partial Difference Equations 145 

Adding these equations we have 

- 16 (00) + 12 (aO) -f 12 (Oa) -f 12 (aO) -f 12 (Oa) 
= 2 (a) + 2 (a) + 2 (a) -f 2 (a) -f 2 [a] -f 2 [a] -f 2 [a] 4- 2 [a] 
+ 4(0) + 4(0) + 4[0]+4[6]. 

Combining this with the homogeneous equation we see that the quantity 
on the right-hand side of the last equation is equal to -f 32 (00) and so 
the quantity (00) is obtained uniquely. 

Similarly, if the sides of the square are x = 3h, y = 3h there are 
16 7i-A-equations and 9 A-equations which may be solved by first eliminating 
the quantities which do not occur in the /^-equations. We have then to 
solve 9 linear equations in order to obtain the remaining quantities, but 
these 9 equations may be treated in exactly the same way as the previous 
set of 9 linear equations, quantities being eliminated which do not occur 
in the central equation. In this way a value is finally found for (00). 

A similar method may be used for a more general type of square net- 
work or lattice. Let the four points (x -f- h, y), (x h, y), (x, y ,+ /*) 
(x, y h) be called the neighbours of the point (x, y) and let the lattice L 
consist of interior points P, each of which has four neighbours belonging 
to the lattice, and boundary points Q, each of which has at least one 
neighbour belonging to the lattice and at least one neighbour which does 
not belong to the lattice. A chain of lattice points A lt A 2 , ... A n+1 is said 
to be connected when ^4 S+1 is one of the neighbours of A s for each value 
of ^ in the series 1, 2, ... n. A lattice L is said to be connected when any 
two of its points belong to a connected chain of lattice points, whether the 
two points are interior points or boundary points. The lattice has a simple 
boundary when any two boundary points belong to a chain for which no 
two consecutive points are internal points and no internal point P is 
consecutive to two boundary points having the same x or the same y as P. 

The solubility of the set of linear equations represented by the equation 

u x -x -f u y y = (A) 

for such a lattice may be inferred from the fact that this set of linear 
equations is associated with a certain quadratic form 

h 2 2 (u x * f uj) 9 

where the summation extends over all the lattice points, and a difference 
quotient associated with a boundary point is regarded as zero when a point 
not belonging to the lattice would be needed for its definition. This sum- 
mation can, by the so-called Green's formula, be expressed in the form 

- WJ^u(u x2 + u v y) - h^uR(u), (B) 

P Q 

where the boundary expression R (u) associated with a boundary point U Q 
is defined by the equation 

hR (U Q ) u -f u 2 -h ... u s &UQ, 



146 Applications of the Integral Theorems of Gauss and Stokes 

where u^u^, ... U B are the 8 neighbouring points of U Q (s < 3). Since 
u xx -f u v ? = the quadratic form can be expressed in terms of boundary 
values. If there were two solutions of the partial difference equation with 
the same boundary values, the foregoing identity could be applied to their 
difference u v y and since the boundary values of u v are all zero the 
identity would give the relation 

2 [(* - *.) + (u, - v y y] = o, 

which implies that u x v x = 0, u y v y = for all points (x, y) of the 
lattice ; consequently, since u v is zero on the boundary it must be zero 
throughout the lattice. 

There is another identity 

= A 2 2 (vu xx -f vUyj uv xx uVyy) + h 2 [vR (u) uR (v)] 

P Q 

which, when applied to the case in which u xx -f u y y = and the boundary^ 
value of v is zero, gives 

2 [(u x + v x )* + (u y + v y )*] = - 2 (u + v) (v xx + v vy ) - h~ l S (u) [R (v) + R (u)] 

P Q 

= - A- 1 S [vR (u) - uR (v)] + 2 (v x * + v v 2 + u x * + u y 2 ) 

Q 

+ A- 1 2 uR (u) - h- 1 2 [uR (u) + uR (v)] 

Q Q 

= S (v. + v y 2 + u x * + u y *) 

> S (uj + u y *), 

the transformations being made with the aid of Green's formula (B). 

This equation shows that the solution of u xx -p u yy = and the pre- 
scribed boundary condition gives the least possible value to the quadratic 
form. The system of linear equations u xx -f u yy = can, indeed, be ob- 
tained by writing down the conditions that the quadratic form should be 
a minimum when the boundary values of u are assigned. 

With a change of notation the quadratic form may be written in the 
form N N N 

2 2 c mn u m u n - 2 2 a n u n + 6, 

ml n 1 n 1 

where the quadratic form is never negative. The corresponding set of linear 

equations 
H -f c ia w 2 -f- ... C IN U N - a 1? 



has a determinant | c mn | which is not zero and so can be solved. 

For the sake of illustration we consider a lattice in which the internal 
lattice points are represented in the diagram by the corresponding values 



Associated Quadratic Form 147 

of the variable u and the boundary lattice points by corresponding values 
denoted by t/s. 



The quadratic form is in this case 
K-*g 2 +K-^i) 2 +K-^ 

to - *>e) 2 4- K - *4>) 2 + (o - ^io) 2 + K - ^) 2 + to - ^i) 2 + to - t> 9 ) 2 , 

(t>3 - ^2) 2 + K - ^8) 2 4- (K, ~ ^4) 2 + (*4 ~ "5) 2 > 

and it is easy to see that the equations obtained by differentiating with 
respect to u , u 2 respectively are 

4^j = u + u 2 + ^ -f v 2J 4^2 = ^ + 1*3+^5 + V 3> 

and are of the required type. The quadratic form is, moreover, equal to 
the sum of the quantities 

t^-Wa-VM-tJoJ+M^ 
- <N O - w 2 - u 4 - v 9 ) 4- w 4 (4w 4 -w 3 -i* 5 - v 1 - v B )+u 6 (4w 6 -^2 -^4-^-^4), 

^o (o - w o) 4- Vi (i ~ ^) + v 2 (^2 - ^i) 4- % to ~ ^2) 4- ^4 (^4 - ) 

-f v 6 to - ^s) 4- v? ( v ? - %) 4- v s (v g - u 4 ) + v 9 (w 9 - Ws) + v 10 (VM - ^ )- 



2-34. TAe limiting process^. We assume that 6 is a simply connected 
region in the ^-plane with a boundary F formed of a finite number of arcs 
with continuously turning tangents. If v is an integrable function defined 
within G we shall use the symbol {v} to denote the integral of v over the 
area G and a similar notation will be used for integrals of v over portions 
of G which are denoted by capital letters. 

Let G h be the lattice region associated with the mesh-width h and the 
region G, and let the symbol G h [v] be used to denote the sum of the 
values of v over the lattice points of G. Also let the symbol I\ (v) be used 
for the sum of the values of v over the boundary points which form the 
boundary I\ of G h . This notation will be used also for a portion of G h 
denoted by a capital letter and for the lattice region Q h * belonging to a 
partial region Q* of G. 

Now let / (x, y) be a given function which is continuous (D, 2) in a 
region enclosing G and let u (#, y) be the solution of (A) which takes the 
same value as / (#, y) at the boundary points of G h . We shall prove that 
as h -> the function u h (x, y) converges towards a function u (x y y) which 

f R. Courant, K. Friedrichs and H. Lewy, Math. Ann. vol. c, p. 32 (1928). See also J. le 
Roux, Journ. de Math. (6), vol. x, p. 189 (1914); R G. D. Richardson, Trans. Amer. Math. Soc. 
vol. xvm, p. 489 (1917); H. B. Phillips and N. Wiener, Journ. Math, and Phys. Mass. Inst. Tech. 
vol. n, p. 105 (1923). 

10-2 



148 Applications of the Integral Theorems of Gauss and Stokes 

satisfies the partial differential equation V 2 u = and takes the same value 
as / (#, y) at each of the points of F. We shall further show that for any 
region lying entirely within G the difference quotients of u h of arbitrary 
order tend uniformly towards the corresponding partial derivatives of 
u (x, y). 

In the convergence proof it is convenient to replace the boundary 
condition u f on F by the weaker requirement that 

<$>{(^~/) 2 }-*0 asr->0, 

where S r is that strip of G whose points are at a distance from F smaller 
than r. 

The convergence proof depends on the fact that for any partial region 
G* lying entirely within G, the function u h (x, y) and each of its difference 
quotients remains bounded and uniformly continuous as h -> 0, where 
uniform continuity is given the following meaning : 

There is for any of these functions w h (x, y) a quantity 8 (e), depending 
only on the region and not on h, such that if w h (p) denote the value of the 
function at the point P we have the inequality 

I *V P) - *^ (PI) | < * 

whenever the two lattice points P and P l of the lattice region G h lie in the 
same partial region and are separated by a distance less than 8 (e). 

As soon as the foregoing type of uniform continuity has been established 
we can in a well-known manner f select from our functions u h a partial 
sequence of functions which tend uniformly in any partial region 6?* 
towards a limit function u (x, y} while the difference quotients of u h tend 
uniformly towards that of u (x, y} differential coefficients. The limit func- 
tion then possesses derivatives of order n in any partial region G* of G and 
satisfies V*u = in this region. If we can also show that u satisfies the 
boundary condition we can regard it as the solution of our boundary 
problem for the region. G. Since this solution is uniquely determined, it 
appears then that not only a partial sequence of the functions u h but this 
sequence of functions itself possesses the desired convergence property as 
A-+0. 

The uniform continuity of our quantities may be established by proving 
the following lemmas : 

(1) As h -* the sums h*Q h [u*] and h*G h [u x 2 + u y z ] remain bounded. 

(2) If w = w h satisfies the difference equation (A) at a lattice point 
of G h and if, as h -> the sum h 2 G A * [w 2 ] , extended over a lattice region 
G h * associated with a partial region G* of G, remains bounded, then for 
any fixed partial region 6?** lying entirely within 6?* the sum 



f See for instance, Kellogg's Foundations of Potential Theory, p. 265. The theorem to be 
used is known as Ascoli's theorem; it is discussed in 4-45. 



Inequalities 149 

over the lattice region 6? A ** associated with 6**, likewise remains bounded 
as Ti-> 0. When this is combined with (1) it follows that, because all the 
difference quotients w of the function u h also satisfy the difference equation 
(A), each of the sums h 2 G h * [w 2 ] is bounded. 

(3) From the boundedness of these sums it follows that the difference 
quotients themselves are bounded and uniformly continuous as h -> 0. 

The proof of (1) follows from the fact that the functional values u h are 
themselves bounded. For the greatest (or least) value of the function is 
assumed on the boundary *f and so tends towards a prescribed finite value. 
The boundedness of the sum h 2 G h [u x 2 -f u y 2 ] is an immediate consequence 
of the minimum property of our lattice function which gives in particular 

h*G h [u x 2 + u y 2 ] < WQ K (f x 2 + A 2 ], 

but as h-> the sum on the right tends to G \( J ) -f ( ~ ) , v , which, by 

[\dx/ \vy>) 

hypothesis, exists. 

To prove (2) we consider the sum h 2 Q l [iv x 2 -f w^ + w y 2 -f- w^ 2 ], where 
the summation extends over all the interior points of a square Q t . Now 
Green's formula gives 

h 2 Qi K 2 + ** + Wy 2 + ?V] = 2 (w 2 ) - S (w 2 ), 

1 

where j and 2 are respectively the boundary of Q v and the square 
boundary of the lattice points lying within Sj . 

We now consider a series of concentric squares Q , Q l , ... Q v with the 
boundaries S , S 19 ... 2^ v - Applying our formula to each of these squares 
and observing that we have always 

2h 2 Q [w x 2 + w y 2 ] < Ji 2 Q k [w, 2 -f w 2 2 + w y 2 + w g *] (k > 1), 
we obtain 

2h 2 Q [w x 2 + w y 2 ] < 2 (w 2 ) - 2 (iv 2 ) (0 < k < n). 

A* f 1 k 

Adding n inequalities of this type we obtain 

2nh 2 Q [w x 2 -{- w y 2 ] < S (w 2 ) - S (w 2 ) < S (w 2 ). 

n n 

Summing this inequality from n = 1 to n = N we get 
N 2 h 2 Q [w x 2 -f ?V] < Q N [w 2 ]. 

Diminishing the mesh-width h we can make the squares Q and Q N 
converge towards two fixed squares lying within and having corre- 
sponding sides separated by a distance a. In this process Nh -> a and we 

have independently of the mesh-width 

/> 2 
h 2 Q [w x 2 ^w v 2 ]< a Q N [w 2 ]. 

Uf 

t On account of equation (A) the value of u h at an internal point is the mean of the values 
at the four neighbouring points and so cannot be greater than all of them, consequently the greatest 
value of M* cannot occur at an internal point. 



150 Applications of the Integral Theorems of Gauss and Stokes 

With a sufficiently small value of h this inequality holds with another 
constant a for any two partial regions of G, one of which lies entirely 
within the other. Hence the surmise in (2) is proved. 

To prove that u h and all its partial difference quotients w remain 
bounded and uniformly continuous as h -> we consider a rectangle R 
with corners P , Q QJ P, Q and with sides P Q Q , PQ which are x-linesf of 
length a. Denoting these lines by the symbols X Q , X respectively we start 

from the formula p 7 ir / v , 7 9 r r -, 
K^O - w p * = hX (w x ) + h*R [w xy ] 

and the inequality 

| W Q Q _ W P Q | < hX (| w x |) + h*R [| w xy |] ...... (C) 

which is a consequence of it. We now let X vary continuously between an 
initial position X l at a distance 6 from X Q and a final position X 2 at a 
distance 26 from X and sum the (b/h) -f 1 inequalities (C) associated with 
X'a which pass through lattice points. We thus obtain the inequality 



where the summations on the right are extended over the whole rectangle 
P Q P 2 Q 2 . By Schwarz's inequality it then follows that 

| w p - u#> | < (2a/6)* (A 2 # 2 [w x 2 ])* -f (2a6)i (& 2 # 2 [^ xv 2 ])*. 

Since, by hypothesis, the sums which occur heVe multiplied by A 2 
remain bounded, it follows that as a -> the difference | w p <> wA | -^ 
independently of the mesh- width, since for each partial region G* of 6? 
the quantity b can be held fixed. Consequently, the uniform continuity of 
w = w h is proved for the ^-direction. Similarly, it holds for the t/-direction 
and so also for any partial region (7* of G. The boundedness of the function 
w h in G* finally follows from its uniform continuity and the boundedness 
of h*G*[w h *]. 

By this proof we establish the existence of a partial sequence of 
functions u h which converge towards a limit function u (x, y) and are, 
indeed, continuous, together with all their difference quotients, in the 
sense already explained, for each inner partial region of G. This limit 
function u (x, y) is thus continuous (Z>, n) throughout (?, where n is 
arbitrary, tod it satisfies the potential equation 



In order to prove that the solution fulfils the boundary condition 
formulated above we shall first of all establish the inequality 

h 2 S r ,h [v 2 ] < ArWS fth [v x 2 -f V] + Brhr h (v 2 ), ...... (D) 

where S f)h is that part of the lattice region G h which lies within the 

t This term is used here to denote lines parallel to the axis of x. 



Properties of the Limit Function 151 

boundary strip S r , which is bounded by F and another curve F r . The 
constants A, B depend only on the region and not on the function v or 
the mesh-width h. 

To do this we divide the boundary F of G into a finite number of pieces 
for which the angle of the tangent with either the x or t/-axis is greater 
than 30. Let y, for instance, be a piece of F which is sufficiently steep 
(in the above sense) relative to the #-axis. The x-lines through the end- 
points of the piece y cut out on F r a piece y r and together with y and y r 
enclose a piece s r of the boundary strip S r . We use the symbol s ft h to denote 
the portion of G h contained in s r and denote the associated portion of the 
boundary F A by y h . 

We now imagine an o?-line to be drawn through a lattice point P h of 
S r th . Let it meet the boundary y h in a point P h . The portion of this 
#-line X h which lies in s r th we call p r th . Its length is certainly smaller 
than cr, where the constant c depends only on the smallest angle of in- 
clination of a tangent of y to the #-axis. __ 

Now between the values of v at P h and P h we have the relation 
v p h ^ v p h hx h (v x ). 

Squaring both sides and applying Schwarz's inequality, we obtain 
(v f *)* < 2 (iA) 2 + 2crhp Tfh (v x 2 ). 

Summing with respect to P h in the ^-direction, we get 



Summing again in the ^/-direction we obtain the relation 
hs r , h [v*] < 2crT h (v?*)* + 2c*r*S r , h [v,*]. 

Writing down the inequalities associated with the other portions of F 
and adding all the inequalities together we obtain the desired inequality 
(D). 

By similar reasoning we can also establish the inequality 

h*Q h 0*] < Cl hr h (v*) -f c 2 h*G h (V + V] 

in which the constants c ls c 2 depend only on the region G and not on the 
mesh division. 

We now put v h = u h f h so that v h = on F A . 

Then, since h 2 G h [v x 2 -f v y 2 ] remains bounded as h -> 0, we obtain from 

(D) (h*/r)S rtk [v*]<Kr, ...... (E) 

where K is a constant which does not depend on the function v or the mesh- 
width. Extending the sum on the left to the difference S^ h S pfh of two 
boundary strips, the inequality (E) still holds with the same constant K 
and we can pass to the limit h -* 0. 
From the inequality (D) we then get 

(l/r)S[v*]<Kr, 



152 Applications of the Integral Theorems of Gauss and Stokes 

where 8 S r S fl and v u /. Now letting /> -> 0, we obtain the 
inequality ^ ^ ^ ^ (1/jp) ^ [(u _ /)2] ^^ ^ 

which signifies that the limit function satisfies the prescribed boundary 
condition. 

2-41. The derivation of physical equations from a variational principle. 
\ concise expression may be given to the principles from which an equation 
or set of equations is derived by using the ideas of the " Calculus of 
Variations*." This expression is useful for several purposes. In the first 
place a few methods are now available for the direct solution of problems 
in the "Calculus of Variations" and these can sometimes be used with 
advantage when the differential equations are hard to solve. Secondly, 
when an integral's first variation furnishes the desired physical equations 
the expression under the integral sign may be used with advantage to 
obtain a transformation of the physical equations to a new set of co- 
ordinates, for the transformation of the integral is generally much easier 
than the transformation of the differential equations and the transformed 
equations can generally be derived from the transformed integral by the 
methods of the "Calculus of Variations," that is, by the Eulerian rule. 

To illustrate the method we consider the variation of the integral 

r i f ff [f3 v \ 2 i sv \ 2 /2F 

/== 6 a 'I + "a -) + br 

2JJJ [\dxj \dy/ \dz 

when the dependent variable V is alone varied and its variation is chosen 
so that it vanishes on the boundary of the region of integration. We have 



Now by a fundamental property of the signs of variation and differentia- 
tion a I/ a 

^ V ~ J7 

8-x Q- (SF), etc. 

dx 8x v h 

Hence 87 - f I f \~ j^ (8V) + ...1 dxdydz 



V~dS- \8V.V 2 Vdxdydz. 

dn J J J y 

The surface integral vanishes because 8F = on the boundary, conse- 
quently the first variation 87 vanishes altogether if F satisfies everywhere 

the differential equation .-.^^ 

^ V 2 !/ = 0. 

The condition 8 F = on the boundary means that as far as the possible 
variations of F are concerned F is specified on the boundary. It is easily 

* The reader may obtain a clear grasp of the fundamental ideas from the monograph of 
G. A. Bliss, "Calculus of Variations," The Carus Mathematical Monographs (1925). 



Variational Principles 153 

seen that a function V with the specified boundary values gives a smaller 
value of / when it is a solution of V 2 V = 0, regular within the region, than 
if it is any other regular function having the assigned values on the 
boundary. 

In the foregoing analysis it is tacitly assumed that V 2 V exists and is 
such that the transformation from the volume integral to the surface 
integral is valid. If V is assumed to be continuous (Z>, 2) there is no difficulty 
but, as Du Bois-Reymond pointed out*, it is not evident that a function V 
which makes 87 = does have second derivatives. This difficulty, which 
has been emphasised by Hadamardf and LichtensteinJ, has been partly 
overcome by the work of Haar . There are in fact some sufficient conditions 
which indicate when the derivation of the differential equation of a varia- 
tion problem is permissible. 

For the corresponding variation problem in one dimension there is 
a very simple lemma which leads immediately to the desired result. The 
variation problem is 

8 



where x l and x 2 are constants and 8 V is supposed to be zero for x = x\ 
and for x x 2 . 

dV 
Writing - = M , 8 V = U we have to show that if 



- 

for all admissible functions U which satisfy the conditions 

U(x,)= U(x 2 ) = ...... (B) 

then M is a constant (Du Bois-Reymond's Lemma). 

To prove the lemma we consider the particular function 

U(x) = (x 2 - x) \ X M (f) dg - (x - x,) PMtf) dg, 

J Xt JX 

which satisfies (B) and gives at any point x where M (x) is continuous 



- c], say. 

* P. du Bois-Reymond, Math. Ann. vol. xv, pp. 283, 564 (1879). 

f J. Hadamard, Comptes Rendus, vol. CXLIV, p. 1092 (1907). 

J L. Lichtenstein, Math Ann. vol. LXEX, p. 514 (1910). 

A. Haar, Journ. fur Math. Bd. CXLIX, S. 1 (1919); Szeged Acta, t. in, p. 224 (1927). Haar 
shows that in the case of a two-dimensional variation problem the equation 87 = leads to a pair 
of simultaneous equations of the first order in which there is an auxiliary function W whose 
elimination would lead to the Eulerian differential equation if the necessary differentiations 
could be performed. Many inferences may, however, be derived directly from the simultaneous 
equations without an appeal to the Eulerian equation. 



154 Applications of the Integral Theorems of Gauss and Stokes 

We shall now assume that M(x) is continuous bit by bit (piece wise con- 
tinuous) so that this equation holds in the interval (x l , x 2 ) except possibly 

at a finite number of points. The functions M (x), -j- are then undoubtedly 

(tx 

integrable over the range (x l9 x 2 ) and, on account of the end conditions (B) 
satisfied by U(x), we may write (A) in the form 



With the value adopted for U this equation becomes 

cx t 

[M (x) c] 2 dx = 0, 

dM d 2 V 

and implies that M (x) c, hence -7- = and -v-z = 0. 

a# dx 2 

An extension of this analysis to the three-dimensional case is difficult. 
To avoid this difficulty it is customary* to limit the variation problem and 
to consider only functions that are continuous (D, 2) throughout the region 
of integration. The function V and the comparison function V + U are 
supposed to belong to the field of functions with the foregoing property. 
The problem is to find, if possible, a function V of the field such that 81 
is zero whenever U belongs to the field and is zero on the boundary of the 
region of integration. 

Even when the problem is presented in this restricted form a lemma is 
needed to show that V necessarily satisfies the differential equation. We 
have, in fact, to show that if 

U.V 2 V dxdydz^ 0, 

for all admissible functions U, then V 2 V = 0. 

The nature of the proof may be made clear by considering the one- 
dimensional case. We then have the equations 

<f) (x) U (x) dx = 0, 

and the conditions : 

U (x) is continuous (Z>, 2), <f>(x) is continuous in (x l9 x 2 ). 
Since U (x) is otherwise arbitrary we may choose the particular function 
U (x) = (x a) 4 (b #) 4 x l < a < x < 6 < x 2 

= otherwise. 

If <f>(x) were not zero throughout the interval (x lf x 2 ) it would have a 
definite sign (positive, say) in some interval (a, 6) contained within (x l9 # 2 ), 

* See, for instance, Hilbert-Courant, Methoden der Mathematischen Physik, vol. I, p. 165. 



Fundamental Lemmas 155 

but this is impossible because with the above form of U(x) the integral 
<f> (x) U(x)dx is positive. 

J X\ i 

To extend this lemma to the three-dimensional problem it is sufficient 
to consider a function U(x, y, z) which has a form such as 

within a small cube with (a lf a 2 , a 3 ), (b lt 6 2 > ^3) as ends of a diagonal, the 
value of U outside the cube being zero. 

In this way it can be shown that a field function V for which 81 
is necessarily a solution of V 2 V = 0. The foregoing analysis does not prove, 
however, that such a function exists. 

Similar analysis may be used to derive the equation V 2 <f> + k*(f> = from 
a variational principle in which 

8 If \Ldxdydz = 0, 



When the potential <f> is of the form - cos kr the volume integral is 
finite although the integral k 2 <f> 2 is not*. 

EXAMPLES 

i. if /ass f/T(li) 8 " 

the equation 87 = may be satisfied by making V f(x + y) + g(x y) where / and g 
have first derivatives but not necessarily second derivatives. [Hadamard.] 

2. The variation problem 

8 F(V X , V y , x, y)dxdy = 

leads to the simultaneous equations 

W- dF W -~~ 
x ~dV y ' v ~ BV X ' 

the suffixes x, y denoting differentiations with respect to these variables. [A. Haar.] 

242. The general Eulerian rule. To formulate the general rule for 
finding the equations which express that the first variation of an integral 
is zero we consider the variation of an integral 

f f f 
/ = ... L dx l dx 2 . . . dx n , 

where L is a function of certain quantities and their derivatives. For 

, * See, for instance, the remarks made by J. Lennard- Jones, Proc. London Math. Soc. vol. xx, 
p. 347 (1922). 



156 Applications of the Integral Theorems of Gauss and Stokes 

brevity we use Suffixes 1, 2, etc. to denote derivatives with respect to 
#!, x 2 , etc. If there are m quantities u, v, w, ... which are varied inde- 
pendently except for certain conditions at the boundary of the region of 
integration, there are m Eulerian equations which are all of type 



o = _ - - - - ~ e 

du S ,idx 3 \duj 2l a ^it^i dx a dx t \du st ) at 
1 " " n 3 3 / SL 

~ 3 ! r ?t .?! aavaar.a*; l 

these are often called the Euler-Lagrange differential equations, but for 
brevity we shall call them simply the Eulerian equations. 

If /j , 1 2 , . . . l n are the direction cosines of the normal to an element of 
the boundary, the boundary conditions are of types 



, 
' 



rdL i a / a/A i a / a/A i 
ldu. 2! 2i aTA e "a^J + 3! 2j a*;aa:A e "'9rJ "T 



r-l 



0= S 

a -! 

There are m boundary conditions of the first type, mn boundary con- 
ditions of the second type, ^mn(n 1) boundary conditions of the third 
type, and so on. In these equations the coefficients $ st , rst are constants 
which are defined as follows : 

8t - 1 S ^ t, 

= 2 8= t, 

e rst = 1 r ^ s ^ t, 
= 2 r = s ^ t, 
= 6 r = s ^ t. 

The equations (A) are obtained by subjecting the integral to repeated 
integrations by parts until one part of the integral is an integral over the 
boundary and the other part is of type 

||... l[USu+ V8v+ WSw+ ...]dx, ... dx n . 
The equations ^ ^ y _ ^ w = Q? 

are then the Eulerian differential equations*, while the boundary integral is 

(dS [USu + S U t Su t + L U n Su rt + ...], 



and the boundary conditions are 

f/ = 0, f/ ( =0 (=1, 2,...), f/ rt =0 (r= 1,2, ...;<= 1,2,...). 

* For general properties of the Eulerian equations see Ex. 2, p. 183, and the remarks at the 
end of the chapter. 



The Eulerian Equations 157 

Typical integrations by parts are 

^ d [su 1 -ou a ( SL \ 

a r ail j) r, l_/^\] s ^ 2 /^\ 

a 2 /dL 



r s 3L] a r a a /3L\i , a 

U I i \ ~ ^ \ u ^ - \ ^ }\ + OU ~ 

[_ ^ U u\ v x i L dx 2 \du u / \ dx^ 



~ a __ a r\ 3//1 a r a /a//\~] ~ d* /dL' 

The reason for the introduction of the factors t st is now apparent. 

When L depends only on a single quantity u and its first derivatives 
the Eulerian equation is of the second order. The variation problem is 
then said to be regular when this partial differential equation is of elliptic 
type. The distinction between regular and irregular variation problems 
becomes apparent when terms involving the square of 8u are retained and 
the sign of the sum of these terms is investigated (Legendre's rule). 

When a variation problem is irregular it is not certain that the boundary 
conditions suggested by the variation pioblem will be equivalent to those 
which are indicated by physical considerations. 

For a physically correct variation problem a direct method of solution 
is often advantageous. The well-known method of Rayleigh and Ritz 
is essentially a method of approximation in which the unknown function 
is approximated by a finite series of functions, each of which satisfies the 
specified boundary conditions. The coefficients in the series are chosen 
so as to make 81 = when each coefficient is varied. The problem is thus 
reduced to an algebraic problem. 

2-431. The transformation of physical equations. In searching for 
simple solutions of the partial differential equations of physics it is often 
useful to transform the equations to a new set of co-ordinates and to look 
for solutions which are simple functions of these co-ordinates. The necessary 
transformations can be made without difficulty by the rules of tensor 
analysis and the absolute calculus, but sometimes they may be obtained 
very conveniently by transforming to the new co-ordinates the integral 
which occurs in a variational problem from which they are derived. The 
principle which is used here is that the Eulerian equations which are 
derived from the transformed integral must be equivalent to the Eulerian 
equations which were derived from the original integral because each set 
of equations means the same thing, namely, that the first variation of the 
integral is zero. A formal proof of the general theorem of the covariance 
of the Eulerian equations can, of course, be given *, but in this book we shall 

* L. Koschmieder, Math. Zeits. Bd. xxiv, S. 181, Bd. xxv, S. 74 (1926); Hilbert and 
Courant, I.e. p. 193. 



158 Applications of the Integral Theorems of Gauss and Stokes 

regard this property of covariance as a postulate. It is well known, of 
course, that the postulate leads to the Lagrangian equations of motion in 
the simple case when the integral is of type 

Ldt, 

where L = / [q ly q 2 , ... q n \ q ly q 2 , ... q n ] 

= T - F, 

the Lagrangian equations being of type 

dL c 



= , __. , 

dq 3 dt \Sq s J ' 

The quantity T here denotes the kinetic energy and V the potential 
energy. V is a function of the co-ordinates which specify a configuration 
of the dynamical system, while T is a positive quantity which depends on 
both the q's and their rates of change, which are* denoted here by g's. 

In the simple case when 

n n 

m IVV/v A A 

j- = 2j Zj d rs q r q s , 

i i 

n n 
TT l.VV/ n n 

11 

where the coefficients a rs , c rs are constants, the covariance of the equations 



is easily confirmed by considering a linear transformation of type 

Ql ^ 



Qn = Inl9l + Inn9n> 

in which the coefficients l rs are constants. 

The advantage of making a transformation is well illustrated by this 
case, because when the transformation is chosen so that the expressions 
for T and V take the forms 



respectively, the Eulerian equations are simply 

A,Q, + C S Q, = 0, 
and indicate that there are solutions of type 

Q. = a s cos (n,t + ft), (n*A, = C t ) 

where a s and j8, are arbitrary constants. These co-ordinates Q 3 are called 
the normal co-ordinates for the dynamical problem. 

Our object now is to see if there are corresponding sets of co-ordinates 
associated with a partial differential equation. 



Transformation of Eulerian Equations 159 

2-432. To transform Laplace's equation to new co-ordinates , 77, 
such that 



dx* + dy 2 + dz 2 = ad^ 2 + bd^ + cdt* 

where a, 6, c, /, g, h, are functions of , 77, , we use suffixes a:, t/, z to 
denote differentiations with respect to x, y, z and suffixes 1, 2, 3 to denote 
differentiations with respect to f , rj, . We then have 



7 df d^dt, say, 
. '? t) / 



say, 

h - af) = J*F, say. 
Therefore 

2 F 3 + 20V 3 F, + 2H V l F 2 ] 



By Euler's rule 87 = when 

9 / 3L\ 9_ / 9L_\ 8 / 3L \ _ 

a? va v\ ) + ^ V3 F 2 y + a^ va F 3 ) ~ u> 

The new form of Laplace's equation is thus 
nv a 

UV = ^. 



^A I *-* n i -* oy 

c/f C/TJ c/4 

8 



If the original integral is 

P, a + F V 2 + V z 2 - XV*]dxdydz, (A) 

the transformed integral is 

where U = L |AF 2 /J, 

and so the equation V 2 F + AF = 0, 

which is derived from (A) by Euler's rule, transforms into the equation 

DV + XV IJ = 



160 Applications of the Integral Theorems of Gauss and Stokes 
which is derived from (B) by Euler's rule. This shows that V 2 F transforms 

into J.DV, 

where J 2 a h g = 1. 

! & b f \ 

\ 9 f c 

This result was given by Jacobi* with the foregoing derivation. The 
particular case in which 



was worked out by Lam6. The result is that 

2 dV\ 3 / A 8 8F 



This result is of great importance and will be used in the succeeding 
chapters to find potential functions and wave-functions which are simple 
functions of polar co-ordinates, cylindrical co-ordinates and other co- 
ordinates which form an orthogonal system. 
In the special case when 

dx 2 -f dy* + dz* - * 2 (r/ 2 H- d^ -f d 2 ), 
Laplace's equation becomes 

a / dv\ a / dv\ a / sv\ 

^^\ K '^7 }^ ^ \ K -*- H~ ^ K 

3| V 9^ ; Srj\ drjj S 
and implies that /c-F is a solution of 



if K^ is a solution of this equation. 

Inversion is one transformation which satisfies the requirements, for in 
this case * = /**, y = 



The inference is that if F (x, y, z) is a solution of Laplace's equation, 
the functiont , 

1 F x y z 

r \r 2 ' r 2 ' r 
is also a solution. Another transformation which satisfies the requirements 

is , _ ax_ r 2 - a 2 , _ r 2 -f a 2 

^ ~ y~+Tz ' ^ ~ 2 (y"+ " w) ' Z ~ 2i (y + iz) ' 

* Journ. ftir Math. vol. xxxvi, p. 113 (1848). See also J. Larmor, Caw6. Phil. Trans, vol. 
xii, p. 455 (1884); vol. xiv, p. 128 (1886). H. Hilton, Proc. London Math. Soc. (2), vol. xix, 
Records of Proceedings, vii (1921). Some very general transformation formulae are given by 
V. Volterra, Rend. Lined, ser. 4, vol. v, pp. 599, 630 (1889). 

f This result was given by Lord Kelvin in 1845. 



Special Transformations 161 

In this case 

dx'* + dy'* + dz'* = - ^ -- 2 [dx* + <fy 2 

~ 



a 2 1 
w) J 



and we have the result that if F (x, y, z) is a solution of Laplace's equation, 

ax r 2 j--_a 2 r 2 + 

,-+- > 2 (*T-Mz) ' 2i 
is also a solution. 

These two results may be extended to Laplace's equation in a space of 
n dimensions ^ y yy ^ y 

dX} 2 dx 2 2 '" dx n 2 
If F (#!, # 2 , ... x n ) is a solution of this equation, and if 



is also a solution*, and 

(*+ix\~*F\ T *- a2 r2 t 8 ^3_ ^ 1 

1 x "^ 2; L 2 (i + ^2) ' 2t (ij + tij J *! Hh is, ' ' ' ' ^ + ix J 

is a second solution. We shall now use this to obtain Brill's theorem. 
Putting Xj 4- ix 2 = t, x ix 2 = s, the differential equation becomes 



and the result is that if H 2 = a; 3 2 4- ... x n 2 , and if 

jP (s,t,x 3 , ... ar n ) 
is a solution, then 



n 

~ 



is also a solution. Now a particular solution is given by 

8 

v = e u (t) x$, XD ... x n )) 
where U (t, # 3 , # 4 , ... x n ) is a solution of the equation of heat conduction 

dU [d 2 U d 2 U d 2 



which is suitable for a space of n 2 dimensions. The inference is that if 
U is one solution of this equation, the function 



a* ax 3 ax, 
f y T"'*"~7 

* The first result is given by B6cher, Bull. Amer. Math. Soc. vol. ix, p. 459 (1903). 

B II 



162 Applications of the Integral Theorems of Gauss and Stokes 

is a second solution. When U is a constant the theorem gives us the 
particular solution 



which may be regarded as fundamental. 

EXAMPLES 

1. Prove that if 

o - z - ct, p**x + iy t a ~z + ct, b ** x - iy, 
a'-z' + ctf, p-x' + iy', a' = z'-ct', b' - x' - iy , 
the relations a' (la up p) = na 4- wp 4- r -f j3' ( ma 4- v j3 4- g), 
a' (wz -f ft - e) = - tm - rib -f - 0' (va 4- mb - /), 
a' (~ ma + v)8 -f q) - foz -f j -I- A; - 6' (Za ~ w/3 - ^p), 
a' (va -f- mb - f) = ja - M -f + 6' (wa + Z6 - e), 

in which J, m, n, u, v t w,f, g, h, p, q, r, s, e,j, k are arbitrary constants, lead to a relation of type 
dx f * + dy'* -h dz'* - c z dt' z A 2 (dx* -f- dy* -f ^2 2 - 

2. Prove that if z - c* = ^ + (x + iy) 0, 

(z -f d) = * - (x - iy), 

the relations of Ex. 1 give z' ct' $ + 0' (#' -f- iy'), 

0' (*' + <*') = !/>' -(*'-/), 
where X^ ^ w ' ~~ v< t>' + w^' 4- J, 



2*51. jPAe equations for the equilibrium of an isotropic elastic solid. 
Let u, v. w be the components of the small displacement of a particle, 
originally at x, y, z, when a solid body is sliglitly deformed, and let X , Y, Z 
be the components of the body force per unit mass. We consider the 
variation of the integral 

/ = Ldxdydz, 

where L = S W, with 



vY 
25 - (A + 2/i) (u x + v v 4- w z )* -f ^ [K + 



'A and p being positive constants. The quantity 8 may be regarded as the 
strain energy per unit volume, while W is the work done by the body 
forces per unit volume. The densitj 7 /> is supposed to be constant. 

We now wish L to be a minimum subject to the condition that the 



Isotropic Elastic Solid 163 

values of u, v and w are specified at the boundary of the solid. The Eulerian 
equations of the "Calculus of Variations" give 



dz 



where X x = 2pu x + X(u x + v y + w z ), Y z = Z y = ^ (iv y 4- v g ), 
Y y = 2fjLV v 4- A (u x 4- v v 4- w,) 9 Z x = .Y z = ^ (u z + w x )> 
Z z = 2/xtik 4- A (u x 4- v v + ^ 2 ), X v = Y x = /x (t^ + w y ). 

The quantities JT,,., y y , Z z , y a , Z^, X v , are called the six components 
of stress, and the quantities 

e x x*=u x) e vv =v v , e zz =w z , 
e yz = w y + v z , e za . = ft, 4- w; x , e^y == v x 4- u V9 

are called the six components of strain. In terms of these quantities 28 
may be expressed in the form 

28 = X x e xx 4- Y y e vv 4- Z z c + r,e v , 4- Z x e gx 4- ^e^, 
while the relations between the components of stress and strain are 
X x = 2fie xx 4- AA, Y z = Z y = M e yz , 
7 V - 2 M e tfv 4- AA, Z X = X Z = fie ZX9 
Z z = 2/iC 4- AA, JC V = 7 X = ne xy , 
A = u x -\- v y + w z = e xx + e yy 4- e zz . 
The relations may also be written in the form 
Ee xx = X x - v(Y y + Z z ), 
Ee yy = Y v - a (Z z + JQ, 
1? M = Z,-^ cr (X x + Y v ). 

The coelSicient E is Young's modulus, the number a is Poisson's ratio, 
and /A is the modulus of rigidity. The quantity A is the dilatation and 
A the cubical compression. When 

x m = Y V = z z = - ,, y 2 = z x = x y = o, 

we have e xx = e yy - e zz = - jp/(3A 4- 2/i), 



- A = !>/(A + 
hence the quantity k defined by the equation 

k = A + 



164 Applications of the Integral Theorems of Gauss and Stokes 

is called the modulus of compression. The different elastic constants are 
connected by the equations 

F_M?\+2M) A *__J0_ 

A 4- /* ' 2~(A + /*)' 3 - 6(7* 

On account of the equations of equilibrium the expression for 87 may 
be written in the form 

(X x Su + Y x Sv + Z x Sw) + | (X y 8u + Y y bv + Z y Sw) 

r) 1 

4- Sz (X z 8u 4- Y z 8v + Z 3 8w) dxdydz, 
and may be transformed into the surface integral 

[X v Su -f Y v 
where X v = IX x -f- 



4- ^2, . 

The quantities X v , Y V9 Z v are called the components of the surface 
traction across the tangent plane to the surface at a point under con- 
sideration. In many problems of the equilibrium of an elastic solid these 
quantities are specified and the expressions for the displacements are to be 
found. 

The equations of motion of an elastic solid may l3e obtained by re- 

d'^ii d^i) v^ijo 
garding - ^ 2 - , -^ , ~ 2 as the components of an additional body 

force per unit mass. The equations are thus of type 

dX x dX 3X Z d 2 u 



2-52. The equations of motion of an inviscid fluid. Let us consider the 
variation of the integral 

/ = I j \\Ldxdydzdt, 
where L = pl$ + a-+l (u* -f v + w*)l +/(/>), 



<> < , 

and tt=^ + a 3 S v=? + a^, w=-J- + a- ....... (A) 

ox ox cy dy oz oz ^ ' 

Varying the quantities </, a, j3 and p in such a manner that the variations 



Vortex Motion 165 

of </> and /? vanish on a boundary of the region of integration wherever 
particles of fluid cross this boundary, the Eulerian equations give 



- 

~"' 



, d d d ' 9 9 

where - - == ^- + u 2 - + v =-- + ti; ^ . 

a d< ox ofy dz 

If p = />/' (p) / (p) it is readily seen that 

d^ __ I dp dv __ I dp dw __ 1 3p 
~dt =z ~~f>dx' dt = ~~'pdy' dt = ~pdz' 

where f dp + || + | + J (^ + & + l ^2 ) = j^ ^ ....... (E) 

If > is interpreted as the pressure, the last equation is the usual pressure 
equation of hydrodynamics for the case when there are no body forces 
acting. The quantities u, v, w are the component velocities and p is the 
density of the fluid at the point x, y, z. The equation (B) is the equation 
of continuity and the equations (D) the dynamical equations of motion. 
The relation p = />/' (p) f (p) implies that the fluid is a so-called baro- 
tropic fluid in which the density is a function of the pressure. It should 
be noticed that with this expression for the pressure the formula for L 

becomes T ^ /A . 

L = F (t) - p 

when use is made of the relation (E). 

The foregoing analysis is an extension of that given by Clebsch*. The 
fact that L is closely related to the expression for the pressure recalls to 
memory some remarks made by R. Hargreaves| in his paper "A pressure 
integral as a kinetic potential." The equations of hydrodynamics may also 
be obtained by writing 

t + d / t -l(u 2 +v 2 + ">*)}+ f(p)> 

and varying <f>, a, j3, u, v, w and p independently. 

The equations (A) are then obtained by considering the variations of 
u y v and w. These equations give the following expressions for the com- 

ponents of vorticity : ~ ~ ~ , . 

. _ 9^ 9v __ 9 (a, j8) 

f ~ 9i/~9z"9(*/,z)' 
_ du _ dw _ 9 (a, j8) 
** ~ 97 " fa ~~ d(z y x) ' 
dv du 9 (a, j8) 



Crette'a Journ. vol. LVI (1859). t P^iL Mag. vol. xvr, p. 436 (1908). 



166 Applications of the Integral Theorems of Gauss and Stokes 

These equations indicate that a = constant, j9 = constant are the 
equations of a vortex line. Now the equations (C) tell us that a and )3 
remain constant during the motion of a particle of fluid, consequently a 
vortex line moves with the fluid and always contains the same particles. 

It should be noticed that in these variational problems no restrictions 
need be imposed on the small variations 80, 8/3 at a boundary which is 
not crossed by particles of the fluid because the integrated terms, derived 
by the integration by parts, vanish automatically at such a boundary of 
the region of integration on account of the equation which expresses that 
fluid particles once on the boundary remain on the boundary. 

2-53. The equations of vortex motion and Liouville's equation. Let us 
consider the variation of the integral 

(A) 



' o x ' ' 

O " > " n / " \ 

y fa d(x,y) 

the expressions for u and v being chosen so as to satisfy the equation of 
continuity, * to = Q 

dx dy J 

for the two-dimensional motion of an incompressible fluid. 

Varying the integral by giving and s arbitrary variations which vanish 
at the boundary of the region of integration, we obtain the two equations 



dx z dy 2 dx \ dy) dy \ dx/ 
The first of these gives s = g (</r), 

where g (iff) is an arbitrary function which, when the region of integration 
extends to infinity, must be such that the integral / has a meaning. This 
requirement usually means that u, v and s must vanish at infinity. With 
the foregoing expression for s the second equation takes the form 

t + !^+?W?'M==> (B) 

which is no other than Lagrange's fundamental equation for two-dimen- 
sional steady vortex motion. In the special case when g ($) = Ae**, where 
A and h are constants, the equation becomes 



Liouville's Equation 167 

This equation, which also occurs in Richardson's theory ot the space 
charge of electricity round a glowing wire*, has been solved by Liouvillef, 
the complete solution being given by 



where a and r are real functions of x and y defined by the equation 
a + ir = F (x + iy) and F (z) is an arbitrary function. 

Special forms of F which lead to useful results have been found by 
G. W. WalkerJ. In particular, if r* = x 2 + y 2 , there is a solution of type 

-** Mr I/TV f?Vl - - 

and when n = 1 the component velocities are given by the expressions 

2i/ 2x ,. 



, 

A = 2/ah y s = f- r - - - , 

1 ' h(a* + r 2 ) ' 

which are very like those for a line vortex but have the advantage that 
they do not become infinite at the origin. If we write 

ds ds ds 

8 ~ -j* u a~ + v 5~ > 
dt dx dy 

the quantity s may be defined by the equation 

s = a tan" 1 (y/x), 

and has a simple geometrical meaning. The quantity s may also be inter- 
preted as the velocity of an associated point on the circle r = a which is 
the locus of points at which the velocity is a maximum. 

It should be noticed that if we use the variational principle 

3 f \(u 2 + v 2 - ^ 2 ) dxdy = 0, ...... (D) 

the corresponding equation is 

. ...... < E > 



and the solution corresponding to (C) is of type 

~ __ 2y _ __??__ 

U "" "" ^(02^72) ' V " A^a^-T 2 ) ' 

This gives an infinite velocity on the circle r = a. 

* 0. W. Richardson, The Emission of Electricity from hot bodies, Longmans (1921), p. 50. The 
differential equation was formulated explicitly by ]Vf. v. La lie, Jahrbuch d. Radioaktivitat u. 
Elektronik, vol. xv, pp. 205, 257 (1918). 

f LiouviUe's Journal, vol. xvm, p. 71 (1853). 

{ G. W. Walker, Proc. Roy. Soc. London, vol. xci, p. 410 (1915); BoUzmann Festschrift, p. 242 
(1904). 



168 Applications of the Integral Theorews of Gauss and Stokes 

Other solutions of (B) which give infinite velocities have been discussed 
by Brodetsky *. It seems that the variational principle (A) may have the 
advantage over (D) in giving solutions of greater physical interest. It 
should be noticed that if a boundary of the region of integration is a stream- 
line */r = constant, it is not necessary for 8s to be zero on this boundary. 

When the motion is in three dimensions an appropriate variation 
principle is 87 = 0, where 



/ = n\(u*+v* + w*$ 2 ) dxdydz, 

and the upper or lower sign is chosen according as the vortex motion is of 
the first or second type. To satisfy the equation of continuity when the 
fluid is incompressible and the density uniform, we may put 

11 = a (<J ' r) = 9 ((7 ' r) m = d (> r) 4 = 3 (*' a > r) 
8(,2)' d(z,x)' W d(x,y)' d(x,y,zy 

A set of equations of motion is now obtained by varying cr, r and s in 
such a way that their variations vanish on the boundary of the region of 
integration. These equations are 

3_(5,CT, T) = () 

d (x, y, z) 
, Scr da da _ 9 ($, s, cr) 

' 



and are equivalent to the equations 
d d ( s >^ 



which imply that 



These equations give 

du _ 1 3 jp dv _ 1 9j? rfit? _ 1 dp 
dt ~~ p dx ' dt~~ pdy* dt ~~ pdz* 

where the pressure y> is given by the equation 

^ + \ (u 2 -f v 2 + w 2 s 2 ) = constant. 

The equation of continuity may also be derived from the variation 
problem by adopting Lagrange's method of the variable multiplier. In 

* S. Brodetsky, Proceedings of the International Congress for Applied Mechanics, p. 374 (Delft, 
1924). 



Equilibrium of a Soap Film 169 

^\ ^\ ^\ 
this method / is modified by adding A( y +5-^4- ^ ) to the quantity 

within brackets in the integrand. The quantities A, u, v, w are then varied 
independently. It is better, however, to further modify / by an integration 
by parts of the added terms. The variation problem then reduces to the 
type already considered in 2-52. 

2*54. The equilibrium of a soap film. The equilibrium of a soap film 
will be discussed here on the hypothesis that there is a certain type of 
surface energy of mechanical type associated with each element of the 
surface. This energy will be called the tension-energy and will be repre- 
sented by the integral 



!J 



TdS 



taken over the portion of surface under consideration, T being a constant, 
called the surface tension. This constant is not dependent in any way upon 
the shape and size of the film but it does depend upon the temperature. 
It should be emphasised that a soap film must be considered as having .two 
surfaces which are endowed with tension-energy. The tension-energy is not, 
moreover, the only type of surface energy; perhaps it would be better to 
say film energy ; for there is also a type of thermal energy associated with 
the film, and from the thermodynamical point of view it is generally 
necessary to consider the changes of both mechanical and thermal energy 
when the film is stretched. 

For mechanical purposes, however, useful results can be obtained by 
using the hypothesis that when a film stretched across a hole or attached 
to a wire is in equilibrium under the forces of tension alone, the total 
tension-energy is a minimum. 

Assuming, then, as our expression for the total tension -energy E 

E = 2T [ f (1 + z x 2 4- z, 2 )* dxdy, 



the z-co-ordinate of a point on the surface or rim being regarded as a 
function of x and y, the Eulerian equation of the Calculus of Variations 
gives ~ ~ 



This is the differential equation of a minimal surface. 

When the film is subject to a difference of pressure on the two sides 
and the fluid on one side of the film is in a closed vessel whose pressure is 
p l while the pressure on the other side of the film is p 2 , there is pressure- 
energy (p l ~ p 2 ) V associated with the vessel closed by the film, where V 
is the volume of this vessel. Writing V in the form 



170 Applications of the Integral Theorems of Gauss and Stokes 

where F is a constant and w is the perpendicular from the origin to the 
surface element dS, we consider the variation of the integral 



Now wH = z xz x yz y , 

and so the differential equation of the problem is 

= ft - P* + 2 

This differential equation may be interpreted by noting that the co- 
ordinates of a point on the normal at (x, y) are 

f = x-Rz x /H, 7) = y-Rz y /H, 

where R is the distance of the point from (x, y). If now two consecutive 
normals intersect at this point, we have 

= d = dx - Ed (z x /H) y - drj = dy - Rd (z v /H), 
for dR = 0. Expanding in the form 

= dx [l - R A (z,///)] -dyB j- (z v /H), 



and eliminating dx, dy, we obtain as our equation for R 

n i P T^ (<> fff\^. ^ i 
= l - R [dx (Z * IH) + dy ( 

If R l and R 2 are the roots, we have 



The quantities R l and R 2 are called the principal radii of curvature. A 

minimal surface is thus characterised by the equation -^ + p- = and 

/h ^2 
a surface of a soap film subject to a constant pressure-difference on its two 

sides is shaped in accordance with the equation 

-_[---= constant. 

-! -#2 

When the film is subjected to only a smalLdifference of pressure and is 
stretched across a hole in a thin flat plate we can, to a sufficient approxi- 
mation, put H = 1 in this equation. The resulting equation is 



where K is a constant and the boundary condition is z on the rim. 



Torsion Problems and Soap Films 171 

Now the same differential equation and boundary conditions occur in 
a number of physical problems and a soap-film method of solving such 
problems in engineering practice was suggested by Prandtl and has been 
much developed by A. A. Griffith and G. I. Taylor*. The most important 
problems of this type are : 

(1) The torsion of a prism ( Saint- Venant's theory). 

(2) The flow of a viscous liquid under pressure in a straight pipe. 
These problems will now be considered. 

EXAMPLES 

1. The forces acting on the rim of a soap film of tension T are equivalent to a force F 
at the origin and a couple 0. Prove that 



0= f%T[rx (nxds)}, 



where the vector ds denotes a directed element of the rim and the vector n is a unit vector 
along the normal to the surface of the film. Show by transforming these integrals into surface 
integrals that the force and couple are equivalent to a system of normal forces, the force 
normal to the element dS being of magnitude 



2. The surface of a film closing up a vessel of volume V can be regarded as one of a 
family of surfaces for which C^ -f C 2 is (7, a constant. If within a limited region of space there 
is just one surface of this family that can be associated with each point by some uniform rule 
and if S' is another surface through the rim of the hole, e the angle which this surface makes 
at a point (x, y, z) with the surface of the family through this point, the area of the outer 

surface of the film is I I cos c . dS'. Hence show that the area of the new surface is greater 
than that of the film if it encloses the same volume. 

3. If w = z x , t> = z v , q***u* + ifi, 
show that the variation problem 

8 ffo(q) dxdy = 

leads to the partial differential equation 



where c 2 [qQ" (q) - O f (q)] = q 2 G' (q). 

Show also that the two-dimensional adiabatic irrotational flow of a compressible fluid 
leads to an equation of this type for the velocity potential z, the function G (q) being given 
by the equation 

O (q) - [2a 2 + (y - 1) (U* - fW~\ 
where U, a and y are constants. 

* See ch. vn of the Mechanical Properties of Fluids (Blackie & Son, Ltd., 1923). 



172 Applications of the Integral Theorems of Gauss and Stokes 

2-55. The torsion of a prism. Assuming that the material of the prism 
is isotropic, we take the axis of z in the direction of the generators of the 
surface and consider a distortion in which a point (#, y, z) is displaced to 
a new position (% + u, y + v, z + w), where 

u = ryz, v = rzx, w = r</>, 

and </> is a function of x and y to be determined. The constant r is called 
the twist. This distortion is supposed to be produced by terminal couples 
applied in a suitable manner to the end faces. The portion of the surface 
generated by lines parallel to the axis of z (the mantle) is supposed to be 
free from stress. These are the simplifying assumptions of Saint-Venant. 
It is easily seen that 

du ___ dv __ dw _^dv du _ 
dx dy dz dx dy ~~ ' 
du dw /dd> 

p ____4_ __._,_ r 

** dz^dx (dx 
dw dv 

p _L 

vz dy ^ dz ~ 

Hence, if Z x = fie zxy Z y = ^e ys , X x ^ Y v = Z z = X y == 0, 
the equations of equilibrium 

az ? = az y = az az, = 

dz ' 'dz ' dx dy 
show that Z x and Z y are independent of z and that 

9 /dJ> \ d /d(j> \ 

>r U - 2/ + 3 - U + # ) = 0, 

dx \dx J ) dy \dy J 

9 , 9 V A 

d^ + d^- 

The boundary condition of no stress on the mantle gives 

IZ X + mZ v - 0, 

where (I, m, 0) are the direction cosines of the normal to the mantle at 
the point (x, y, z). 

Let us now introduce the function </f conjugate to </>, then 
20 __ 3i/r d(f) __ 9i/f 
3x dy ' 9?/ "~ 9x ' 



where r 2 = a: 2 + y 2 . The boundary condition may consequently be written 
in the form ~ ~ 






Torsion of a Prism 173 

where % = iff r 2 and ds is a linear element of the cross-section. This 
equation signifies that x is constant over the boundary and so the problem 
may be solved by determining a potential function i/j which is regular 
within the prism and which takes a value differing by a constant from \r- 
on the mantle of the prism. Without loss of generality this constant may 
be taken to be zero if there is only one mantle. 

It should be noticed that the function x satisfies the equation 

3 2 X , 9 2 X__ 2 

3x 2 T dy 2 

and, with the above choice of the constant, is zero on the mantle when 
this is unique. It is often more convenient to work with the function x> 
especially as 



9v 

- -" 

oy 
Since x vanishes on the mantle it is evident that 

\\Z x dxdy = 0, \\Zydxdy = 0. 

The tractions on a cross-section are thus statically equivalent to a 
couple about the axis of z of moment 



M = (.rZ y - yZ,) dxdy = - pr * + y dxdy. 

Integrating by parts we find that 

M = 

The direction of the tangential traction (Z x , Z y ) across the normal 
section of the prism by a plane z = constant is that of the tangent to the 
curve x constant which passes through the point. The curves x = con- 
stant may thus be called "lines of shearing stress." The magnitude of the 

traction is LLT -, where ~- is the derivative of v in a direction normal to 
r 3n on A 

the line of shearing stress. 

In the case of a circular prism 

x = i (a 2 - r 2 ), 
and* in the case of an elliptic prism 



where a and b are the semi-axes of the ellipse and 



174 Applications of the Integral Theorems of Gauss and Stokes 

2-56. Flow of a viscous liquid along a straight tube. Consider the 
motion of the portion contained between the cross-sections z = z l and 
z == z 1 -f h. If A is the area of the cross-section and /> the density of the 
fluid, the equation of motion is 

M* ^ = -4 (ft -A)- D, 

where p l and p 2 are the pressures at the two sections and D is the total 
frictional drag at the curved surface of the tube. If u is the velocity of 
flow in the direction of the axis of z, u will be independent of z if the fluid 
is incompressible and so we may write 

u = u(x, y, t). 

We now introduce the hypothesis that there is a constant coefficient 
of viscosity p, such that 



where ~ denotes a differentiation in the direction of the normal to the 
on 

surface of the tube. Transforming the surface integral into a volume 
integral, we have the equation of motion 

A ,du Al v , ffr /9 2 ^ 3^\ 7 7 

P Ah -ft = A (ft - ft) + J J V ^ 2 + gp) dxdy. 

Since h is arbitrary this may be written in the form 
Su dp 

- 



When the motion is steady this equation takes the form 



where 2Jf = - >p and can be regarded as a constant, because u is in- 

dependent of z. This is the equation used by Stokes and Boussinesq. 
In the case of an elliptic tube 



where 

For an annular tube bounded by the cylinders r = a, r = b we may 

tt = JA: (a 2 - r 2 ) + JJP (6 2 - a 2 ) log (6/r). 
The total flux $ is in this case 

n _ f 6 , wJfaf . (s 2 - I) 2 ) , ,. . 

- 2w J a urdr - - 4 - r ~ log . r (5 = 6/a) 



Rectilinear Viscous Flow 175 

and so the average velocity is 



4 - 

If there is no pressure gradient the equation of variable flow is 

du 



where v = /x/p. This equation is the same as the equation of the conduction 
of heat in two dimensions. The fluid may be supposed in particular to lie 
above a plane z = which has a prescribed motion, or to lie between two 
parallel planes with prescribed motions parallel to their surfaces. 

The simple type of steady motion of a viscous fluid which is given by 
the equation (I) does not always occur in practice. The experiments of 
Osborne Reynolds, Stanton and others have shown that when a viscous 
fluid flows through a straight pipe of circular section there is a certain 
critical velocity (which is not very definite) above which the flow becomes 
irregular or turbulent and is in no sense steady. From dimensional reason- 
ing it has been found advantageous to replace the idea of a critical velocity 
by that of a critical dimensionless quantity or Reynolds number formed 
from a velocity, a length and the kinematic viscosity v of the fluid. In the 
case of flow through a pipe the velocity V may be taken to be the mean 
velocity over the cross-section, the length, the diameter of the pipe (d). 
For steady "laminar flow" the ratio Vd/v must not exceed about 2300. 

In the case of the motion of air past a sphere a similar Reynolds 
number may be defined in which d is the diameter of the sphere. In order 
that the drag may be proportional to the velocity V the ratio Vd/v must 
be very small. 

EXAMPLES 

1. In viscous flow between parallel planes x = a the velocity is given by an equation 



where c is the maximum velocity. Prove that the mean velocity is two-thirds of the 
maximum. 

2. In a screw velocity pump the motion of the fluid is roughly comparable with that of 
a viscous liquid between two parallel planes one of which moves parallel to the other and 
drags the fluid along, although there is a pressure gradient resisting the flow. Calculate 
the efficiency of the pump and find when it .is greatest. 

Work out the distribution of velocity and the efficiency when the machine acts as a 
motor, that is, when the fluid is driven by the pressure and causes the motion of the upper 
plate. 

[Rowell and Finlayson, Engineering, vol. cxxvi. p. 249 (1928).] 

3. The Eulerian equation associated with the variation problem 



18 



176 Applicatidns of the Integral Theorems of Gauss and Stokes 

L. Lichtenstein [Math. Ann. vol. LXIX, p. 514, 1910] has shown that when f(x, y) is merely 
continuous there may be a function u which makes 8/ = and does not satisfy the Eulerian 

equation. 

2*57. The vibration of a membrane. Let T be the tension of the 
membrane in the state of equilibrium and w the small lateral displacement 
of a point of the membrane from the plane in which the membrane is 
situated when in a state of equilibrium, the vibrations which will be con- 
sidered are supposed to be so small that any change in area produced by 
the deflections w does not produce any appreciable percentage variation 
of T. The quantity T is thus treated as constant and the potential energy 

, ( Sw \* , /3w\ 2 l* , , 
+ ( *-) + ( * dxdy 

\dxj \dyj J y 

is replaced by the approximate expression 



Let pdxdy be the mass of the element dxdy. The equation of motion 
of the membrane will be obtained by considering the variation of 



where 

Iff /dw\ 2 , j r7 
IS = - \\ pi - I dxdy, V - 

The integral to be varied is thus 

= 2jJJ L^ \dt) 

where w -= on the boundary curve for all values of t. The Eulerian 
equation of the Calculus of Variations gives 



where c 2 = T/p. 

This is the equation of a vibrating membrane. The equation occurs also 
in electromagnetic theory and in the theory of sound. In the case when 

w is of the fotm . . ^ 

w == sin Ky . v (x, t), 

the function v satisfies the equation 

^v . 

ri ~ K v 

1x 2 

which is of the same form as the equation of telegraphy. 

It should be noticed that a corresponding variation principle 



m m , , , , 

o- - T \i~) - y hr) \dxdydzdt=*Q 

SxJ \dyj \3zjj y 



The Equation of Vibrations 111 

gives rise to the familiar wave-equation 

d 2 w 



which governs the propagation of sound in a uniform medium and the 
propagation of electromagnetic waves. A function w which satisfies this 
equation is called a wave-function. 

Love has shown* that the equation (A) occurs in the theory of the 
propagation of a simple type of elastic wave. 

Taking the positive direction of the axis of z upwards and the axis of 
x in the direction of propagation, we assume that the transverse displace- 
ment v is given by the equation 

v = Y (z) cos (pt fx). 

The components of stress across an area perpendicular to the axis of 
y are . 

V v Y - n v - 

x ^^dx' - z *~ u > z ~^dz 

respectively and so the equation of motion 



*v dY, dY, , dY 



. 



(B) 



p 'dt* " 
takes the form 



When p and JJL are constants this is the same as the equation of a 
vibrating membrane, but when p and //. are functions of z the equation is 
of a type which has been considered by Meissnerf. 

Transverse waves of this type have been called by JeffreysJ "Love 
waves," they are of some interest in connection with the interpretation of 
the surface waves which are observed after an earthquake. 

It may be mentioned that the general equation (B) may be obtained 
by considering the variation of the integral 

, Iff IT i Sv 

J -2lJJKai 

and an extension can be made to the case in which p and JJL are functions 
of x, z and t. 

2-58. The electromagnetic equations. Consider the variation of the 
integral 

n\\ Ldxdydzdt, 

* Some Problems of Qeodynamics, p. 160 (Cambridge University Press, 1911). 

t Proceedings of the Second International Congress for Applied Mathematics {Zurich, 1926). 

{ The Earth, p. 165 (Cambridge University Press, 1924). 



178 Applications of the Integral Theorems of Gauss and Stokes 
where 2L = H x * + H v * + H t * - E* - E v * - E*, 



and 

H 



V 



-fc--fa, ^---g--^, ( A ) 

MV __ ?4* F = - ~ A * 
" dx dy 9 * dt 

If the variations of A x , A V) A z and vanish at the boundary of the 
region of integration, the Eulerian equations give 

~dy"~"dz ^"dT 9 

~dz~~ 8^ r== "aT 5 

~dx ~ ~dy^~df 9 

dx dy dz 
In vector notation these equations may be written 

curlH=-~, divE=0, (B) 

and equations (A) take the form 

H=curlA, E-~-V*. (C) 

ot 

These equations imply that 

!=-??, divH=0. (D) 



The two sets of equations (B) and (D) are the well-known equations of 
Maxwell for the propagation of electromagnetic waves in the ether; the 
unit of time has, however, been chosen so that the velocity of light is 
represented by unity. The foregoing analysis is due essentially to Larmor. 

Writing Q = H + iE, the two sets of equations may be combined 
into the single set of equations 

divQ=0. ...... (E) 



=-, 
ot 

By analogy with (C) we may seek a solution for which 

Q=-icurlL=-~ ? ~VA. ...... (F) 

The relations between L and A may be satisfied by writing 



, --, ...... (G) 



Electromagnetic Equations 179 

where G is a complex vector of type T + tH, while T and II are real ' Hertzian 
vectors ' whose components all satisfy the wave-equation 



When we differentiate to find an expression for Q in terms of G and K 
the terms involving K cancel and we find the Righi-Whittaker formulae 



H = curl(^curir-f ^), \ (I) 

E = curl (curl II - g 
If L = B + iA, A = Y -f- iO, where A, B, O and X F are real, we have 



...... (J) 

E=-curlB=- 8 - V<D, 

VI 

curir, B-- 



O = - div n, T = - div T, J 

where A, B, O and Y are wave-functions which are connected by the 
identical relations ^ ^^ 

= 0. ...... (L) 



The corresponding formulae for the case in which the unit of time is 
not chosen so that the velocity of light is unity are obtained from the 
foregoing by writing ct in place of t wherever t occurs. 

If we write Q' = e ie Q, where 8 is a constant, it is evident that the vector 
Q' satisfies the same differential equations as Q and can therefore be used 
to specify an electromagnetic field (E x , H x ) associated with the original 
field (E, H). It will be noticed that the function L' for this associated field 
is not the same as L, for 

L 9 = H' 2 - E"* = (H 2 - E 2 ) cos 20-2(E.H) sin 20. 

Also (W . H') = (H 2 - E*) sin 20 + 2 (E . H ) cos 26. 

There are, however, certain quantities which are the same for the two 
fields. These quantities may be defined as follows : 



o JP u i? 17 n 
to x = & y Jl z & Z H V = U x , 

X X = E X * + H X *-W, \ (M) 

Y, = E V E, + H V H Z = Z V ,\ 



180 Applications of the Integral Theorems of Gauss and Stokes 

It is interesting to note that these quantities may be arranged so as to 
form an orthogonal matrix * 



X 



iS x iS v 
We have, in fact, the relations 



x, 

Y, 
Z 2 

iS. 



w 



where 



T/f/2 C* 2 Q 2 C 2 T 

rr &x &y &z ~ * i 

X x Y x + A V Y y -f* -X z * 3 G x G y = 0, 

V O i V O i V O i /"Y TI/ A 
Aa-Oa- + A V O V + A 2 Z + C^a; W = U, 

I=l(H*- E*)*+ (E.Hy 2 - 



.(N) 



2-59. TAe conservation of energy and momentum in an electromagnetic 
field. It follows from the field equations (B) and (D) that the sixteen com- 
ponents of the orthogonal matrix satisfy the equations 

PY ajr 8X, 8g a _ 

a^ 'dt ~~ J 



dx^ dy 



dx ' 
"^ i 



57_ v ay z 

3y 3z 

rjiy ^^ 

Cjy Ofj^ 

dy dz 



dy 



dt 

d_G_> 
"dt 

dw 



~dz 



= 



= 0. 



.(A) 



Regarding S x , S y , S z for the moment as the components of a vector S 
and using the suffix n to denote the component along the outward-drawn 
normal to a surface element da of a surface or, we have 



div/S.^T 
dW 



(dr dxdydz) 



dt 



dr 



= - ^rfr. 



In this equation the region of integration is supposed to be such that the 
derivatives of 8 and W in which we are interested are continuous functions 
of x, y, z and t. This will certainly be the case if the field vectors and their 
first derivatives are continuous functions of x, y, z and t. 



* H. Minkowski, Gott. Nachr. (1908). 



Conservation of Energy and Momentum 181 

Let us now regard W as the density of electromagnetic energy and S 
as a vector specifying the flow of energy, then the foregoing equation can 
be interpreted to mean that the energy gained or lost by the region en- 
closed by a is entirely accounted for by the flow of energy across the 
boundary. This is simply a statement of the Principle of the Conservation 
of Energy for the electromagnetic energy in the ether. 

The equations involving O x , G v , G z may be regarded as expressing the 
Principle of the Conservation of Momentum. We shall, in fact, regard G x 
as the density of the ^-component of electromagnetic momentum and 
( X x , X V) X z ) as the components of a vector specifying the flow of 
the ^-component of electromagnetic momentum. 

The vector S is generally called Poynting's vector as it was used to 
describe the energy changes by J. H. Poynting in 1884. The vector G was 
introduced into electromagnetic theory by Abraham and Poincare. 

In the case of an electrostatic field 



if there are only volume charges and the first integral is taken over all 
space, for then the surface integral may be taken over a sphere .of infinite 
radius and may be supposed to vanish when the total amount of electricity 
is finite and there is no electricity at infinity. It should be noticed that in 
the present system of units Poisson's equation takes the form 

V 2 </> -f p = 0, 

where p is the density of electricity. When there are charged surfaces an 
integral of type 



must be added to the right-hand side for each charged surface. 

The new expression for the total energy may be written in the form 

U = 



This may be derived from first principles if it is assumed that <f)8e is the 
work done in bringing up a small charge 8e from an infinite distance 
without disturbing other charges. Now suppose that each charge in an 
electrostatic field is built up gradually in this way and that when an 
inventory is taken at any time each carrier of charge has a charge equal 
to A times the final amount and a potential equal to A times the final 



182 Applications of the Integral Theorems of Gauss and Stokes 

potential. A being the same for all carriers. As A increases from A to A -f dX 
the work done on the system is 

dU = 2 (Ac/>) ed\. 
Integrating with respect to A between and 1 we get 

U = iSe</>. 

The carriers mentioned in the proof may be conducting surfaces capable 
(within limits) of holding any amount of electricity. If the carriers are 
taken to be atoms or molecules there is the difficulty that, according to 
experimental evidence, the charge associated with a carrier can only change 
by integral multiples of a certain elementary charge e. For this reason it 
seems preferable to start with the assumption that W represents the density 
of electromagnetic energy. 

On account of the symmetrical relations 

7 2 = Z v ,etc., S x = O,, etc., 
we can supplement the relations (A) by six additional equations of types 



. (yZ x -zY x ) + (yZ v -zY v ) + (yZ z -zY,)- (yG, - zG v ) = 0, 



a (x8 x + tX.) + (xS v + tX v ) + - (x8. + tX.) + (xW - tO.) = 0. 

The equations of the first type may be supposed to express the Principle 
of the Conservation of Angular Momentum. We shall, in fact, regard 
yG z zO y as the density of the ^-component of angular momentum and 
(zY x yZ x , zY v yZ y , zY z yZ z ) as the components of a vector which 
specifies the flow of the ^-component of angular momentum. 

The equations of the second type are not so easily interpreted. We shall, 
however, regard xW tO x as the density of the moment of electromagnetic 
energy with respect to the plane x = 0. This quantity is, in fact, analogous 
to Smo;, a quantity which occurs in the definition of the centre of mass of a 
system of particles. Here and in the relation S G we have an indication 
of Einstein's relation 

(Energy) = (Mass) (square of the velocity of light) 
which is of such importance in the theory of relativity. 

The quantities (xS x -f tX X) xS v -f tX v , xS z -f tX 9 ) will be regarded as 
the components of a vector which specifies the flow of the moment of 
electromagnetic energy with respect to the plane x = 0. The equation may, 
then, be interpreted to mean that there is conservation of the moment 
with respect to the plane x = 0. There is, in fact, a striking analogy with 
the well-known principle that the centre of mass of an isolated mechanical 
system remains fixed or moves uniformly along a straight line*. 

* A. Einstein, Ann. d. Physik (4), Bd. xx, S. 627 (1906); G. Herglotz, ibid. Bd. xxxvi, S. 493 
(1911); E. Bessel Hagen, Math. Ann. Bd. ijcxxiv, S. 268 (1921). 



Conservation of Angular Momentum 183 

EXAMPLES 

1. Prove that when there are no external forces the equations of motion of an incom- 
pressible inviscid fluid of uniform density give the following equations which express the 
principles of the conservation of momentum and angular momentum, the motion being two- 
dimensional : 



- uy) - yp] + g- [pv(vx - uy) + xp] - 0. 

Hence show that the following integrals vanish when the contour of integration does not 
contain any singularities of the flow or any body which limits the flow, the motion being 
steady: 

I p (v -f iu) (id + vm) ds + I p (m + il) ds, 



I p (xv yu) (ul + vm) ds -f I p (xm yl) ds. 



When the contour does contain a body limiting the flow the integrals round the contour 
are equal to corresponding integrals round the contour of the body. 

2. Let u (#! , x 2 , ... x n ) be a function which is to be determined by a variational principle 
8/ =r 0, where 

r r r * 

I = II ... M(x l9 x 29 ...x nf u 9 u lt u 29 ...u n )dx li dx^...dx n 

and u r = - - . Suppose further that 7 is unaltered in value by the continuous group of 
transformations whose infinitesimal transformation is 



(r = 1, 2, ... 



_ n 

and let Bu = Aw 2 w r Ax r , 

r-l 



n dR 

then 08u- 2 ^T, 

r-i a* r 

where 5 r - - f&x r - ^ 8u (r 1, 2, ... w). 

When the function w satisfies the Eulerian equation = the foregoing result gives a set of 
equations of conservation. [E. Noether, Gott. Nachr. p. 238 (1918).] 

3. If n = 2, / = (u^ <w 2 2 -h )3w 2 ) e y * 2 where o# )5 and y are arbitrary constants, we may 
write Aa^ = e a , A# 2 = < 2 > ^ w iv 2 tt w ^re e x and e a are two independent small quantities 
whose squares and products may be neglected. Hence show that the differential equation 
u u = aw 22 + yaw 2 4- /to leads to two equations of conservation 

A {2^1^ + V^} - (A + y) K 2 + aW 2 2 + ^w 2 -f 
J5j {V + aw 2 2 - /3w 2 } = (D 2 -f 



where Z^EE /-, D 2 = = . [E. T. Copson, Proc. ^rfm. Jfefa^. Soc. vol. XLII, p. 61 (1924).] 
dx^ 0X2 



184 Applications of the Integral Theorems of Gauss and Stokes 
2-61. Kirchhoff' s formula. This theorem relates to the equation 

D 2 " + a (x, y, z, t) = 0, ...... (A) 

and to integrals of type 

u =-- I r~ l f (t - r/c) F (X Q , y , z ) dr . 

Let us suppose that throughout a specified region of space and a specified 
interval of time, u and its differential coefficients of the first order are 
continuous functions of x, y, z and t ; let us suppose also that the differential 

d 2 u d 2 u 
coefficients of the second order such as ~ ; , ~ 2 and the quantity a are 

finite and integrable. 

Let Q be any point (x , ?/ , z ) which need not be in the specified region 
of space and consider the function v derived from u by substituting t r/c 
in place of t, r denoting the distance from Q of any point (x, y, z) in the 
specified region. It is easy to verify that v satisfies the partial differential 

equation 

__ 2r f 9 (x dv\ d /y Sv\ 3 /z 



where [a] denotes the function derived from a by substituting t r/c in 
place of t. 

We now multiply the above equation by and integrate it throughout 

a volume lying entirely within the specified region of space. The volume 
integral can then be split into two parts, one of which can be transformed 
immediately into an integral taken over the boundary of this region. Let 
the point Q be outside the region of integration, then we have 

3 /1\ I3v 2dr dv] , f [a] , 
V ^ }- - * --- cf ^ \ dS + dr ' 

[_ dn\rj rdn crdndt] } r 

When Q lies within the region of integration the volume may be sup- 
posed to be bounded externally by a closed siirface S 1 and internally by a 
small closed surface S 2 surrounding the point Q. Passing to the limit by 
contracting 8 2 indefinitely the value of the integral taken over S 2 is 
eventually 



ff f 
- \\\ 

JJ [_ 



wjjiere V Q denotes the value of v at Q and this is the same as the value of 
u at Q. Hence in this case 

fW^ flT a f l \ ldv 2dvdr ']jv 

TtU Q == rfr - Vx - ( - ) -- a - - -^5~\dS. 

Q I r JJ l dn\rj rdn crdtdn] 

dv [du~] dr [dul 
Now 



Kirchhoff's Formula 185 

hence finally we have Kirchhoff's formula* 

f M 7 ff ( r n 8 /1\ 1 [dul 1 dr rSi^l) 7r> 

4rrw Q = LJ dr - N[w] =- (- - U- -- U, dS, 

Q 1 r jj \ L *dn\r) r [dn] crdn[dt]\ 

where a square bracket [/] indicates that the quantity /is to be calculated 
at time t r/c. When the point Q lies outside the region of integration 
the value of the integral is zero instead of U Q . 

When u and a are independent of t the formula becomes 

47TU Q = l-dr \\ \U~- (-} =- , , 

J r ]J(dn \rj rdn) 

and the equation for u is 

V*u + a (x, y, z) = 0. 

If we make the surface S l recede to infinity on all sides the surface 
integrals can in many cases be made to vanish. We may suppose, for 
instance, that in distant regions of space the function u has been zero until 
some definite instant t Q . The time t r/c then always falls below t Q when 
r is sufficiently large and so all the quantities in square brackets vanish. 

rs 

The surface integral also vanishes when u and ~ become zero at infinity 

and tend to zero as r -> oo in such a way that u is of order r~ l and ~- , ~- 
of order r~ 2 . In such cases we have the formula 

477^ dr, (B) 

where the integral is extended over all the regions in which the integrand 
is different from zero. 

If [a] exists only within a number of finite regions which do not extend 
to infinity the function U Q defined by this integral possesses the property 
that U Q -> like r ~ l as r -> oo, r being the distance from the origin of 

co-ordinates, but it is not always true that -~ is of order r ~ 2 . To satisfy 

this condition we may, however, suppose that ^- is zero for values of t 

*"*> i 
less than some value t . Then if r is sufficiently large ^ is zero because 

r/c falls below t . 

Wave-potentials of type (B) are called retarded potentials; the analysis 
shows that they satisfy the equation (A) and that the surface integral 

ffkilfiU- 1 ^!- 1 *:^!! 

JJ ( J dn\rJ r [dn] crdn[dt]\ 



* G.Kirchhoff, Berlin. Sitzungsber. S. 641 (1882); Wied. Ann. Bd. xvni (1883); Qes. AM. Bd. u, 
S. 22. The proof given in the text is due substantially to Beltrami, Rend. Lined (5), t. iv (1895), 
and is given in a paper by A. E. H. Love, Proc. London Math. Soc. (2), vol. i, p. 37 (1^03). An 
extension of Kirchhoff's formula which is applicable to a moving surface has been given recently 
by W. R. Morgans, Phil. Mag. (7), vol. ix, p. 141 (1930). 



186 Applications of the Integral Theorems of Gauss and Stokes 

represents a solution of the wave-equation except for points on the surface 
S, for this integral survives when we put a = 0. It should be noticed, 

however, that when we put a = the quantities [u] , - , ^- become 

those relating to a wave-function u which is supposed in our analysis to 
exist and to satisfy the postulated conditions. When the quantities [u]> 

21 > 12" are c " losen arbitrarily but in such a way that the surface integral 

exists it is not clear from the foregoing analysis that the surface integral 
represents a solution of the wave-equation. If, however, the quantities 

[u] , Ur- L ^r possess continuous second derivatives with respect to the 

time the integrand is a solution of the wave-equation for each point on 
the surface. It can, in fact, be written in the form 



where [u] -/*-, = ? ' ~ ' Now stands f or 



7 a 3 3 

ZQ-+ W =- + 7&=- , 

d# <7J/ C7Z 

where I, m and ?i are constants as far as x, y and z are concerned and each 
term such as ^- -/( -- ) is a solution of the wave-equation; consequently 

OX \Jf \ C/ J 

the whole integrand is a solution of the wave-equation and it follows that 
the surface integral itself is a solution of the wave-equation. 

In the special case when a and u are independent of t we have the result 
that when or satisfies conditions sufficient to ensure the existence and 
finiteness of the second derivatives of V (see 2' 32) the integral 



r 
is a solution of Poisson's equation 

V 2 F -f 47T(T (x, y, z) = 0, 

and the integral U = f [ \u J- (-} - - ^1 dS 

e JJ { dn\rj rdn) 

is a solution of Laplace's equation. 

2-62. Poisson's formula. When^he surface S l is a sphere of radius ct 
with its centre at the point Q, [u] denotes the value of u at time t = and 
Kirchhoff 's formula reduces to Poisson's formula * 



* The details of the transformation are given by A. E. H. Love, Proc. London Math. Soc. (2), 
vol. I, p. 37 (1903). 



Poisson's Formula 187 

where /, g denote the mean values of /, g respectively over the surface of 
a sphere of radius ct having the point (x y y, z) as centre and u is a wave- 
function which satisfies the initial conditions 

u=f(x,y 9 z), ^ =0(z,2/,z), 
when t = 0. 

If we make use of the fact that each of the double integrals in Poisson's 
formula is an even function of t we may obtain the relation* 



^ u(x,y,z,t)dt=f. 
This relation may be written in the more general form 



^J u(x,y,z,8)ds 

1 f ff f2ir 

== j- u (x + CT sin 6 cos <f>, y -f CT sin sin <f>, z -f CT cos 0, ) sin 9ddd<f>. 

47T J J 

When w is independent of the time this equation reduces to Gauss's well- 
known theorem relating to the mean value of a potential function over a 
spherical surface. 

If u (x, y, z } s) is a periodic function of s of period 2r, where T is in- 
dependent of x, y and z, the function on the left-hand side is a solution of 
Laplace's equation, for if 



rt + r 
V = c 2 u(x, y, z, s) ds, 

Jt-T 



we have V 2 F = c 2 Vhids = ^ 9 d9 = 0. 



It then follows that the double integral on the right-hand side is also a 
solution of Laplace's equation. 

If in Poisson's formula the functions / and g are independent of z the 
formula reduces to Parseval's formula for a cylindrical wave-function. 
Since we may write 

c 2 * 2 sin Oddd<f> = da . sec 6 = ct (cH 2 - /> 2 )~* da, 

where da is an element of area in the #y-plane and p the distance of the 
centre of this element from the projection of the centre of the sphere, we 
find that 

277 . u (x, y, *) = I JJ da . (cH* - p 2 )^/ (x 4- f , y 4- ij) 

da . (cW - p 2 )'* g (x + , y + r?), 



where da = d^d-q and the integration extends over the interior of the circle 

2 2 



Cf. Rayleigh's Sound, Appendix. 



188 Applications of the Integral Theorems of Gauss and Stokes 

This formula indicates that the propagation of cylindrical waves as 
specified by the equation G 2 w = is essentially different in character from. 
that of the corresponding spherical waves. In the three-dimensional case 

the value of a wave-function u (x, y, z, t) at a point (#, y, z) at time t is 

fjfj 
completely determined by the values of u and ^- over a concentric sphere 

ot 

of radius cr at time t T. If a disturbance is initially localised within a 
sphere of radius a then at time t the only points at which there is any 
disturbance are those situated between two concentric spheres of radii 
ct -f- a and ct a respectively, for it is only in the case of such points that 
the sphere of radius ct with the point as centre will have a portion of its 
surface within the sphere of radius a. This means that the disturbance 
spreads out as if it were propagated by means of spherical waves travelling 
with velocity c and leaving no residual disturbance as they travel along. 
In the two-dimensional case, on the other hand, the value of u (x, y, t) 

at a point (x, y) at time t is not determined by the values of u and -^- over 

a concentric circle of radius CT at time t T. To find u (x, y, t) we must 

f)tj 
know the values of u and -^- over a series of such circles in Which r varies 

ot 

from zero to some other value r l . If the initial disturbance at time t = is 
located within a circle of radius a, all that we can say is that the disturbance 
at time t is located within a circle of radius ct + a and not simply within 
the region between two concentric circles of radii ct -j- a, ct a respectively. 
Hence as waves travel from the initial region of disturbance with velocity 
c they leave a residual disturbance behind. 

The essential difference between the two cases may be attributed to 
the fact that in the three-dimensional case the wave-function for a source 
is of type r~ l f (t r/c), while in the two-dimensional case it is of type* 



C J Jo L c 

This statement may be given a physical meaning by regarding the wave- 
function as the velocity potential for sound waves in a homogeneous 
atmosphere, a source being a small spherical surface which is pulsating 
uniformly in a radial direction. 

If /(0 = 0, (t<T ) 

= 1, (T l >t>T ) 
= 0, (t>TJ, 

we have / \t -cosh a \ da = 0, (ct < cT Q + />) 

Jo L c J 

- cosh- 1 [c (t - T )/p] , (cT + p < ct < cT l + p) 

- cosh-* [c (t - T )//>] - cosh-i [c (t - Tj/p], 

(cT Q -h p < ct, cTi + p < ct). 

* Cf. H. Lamb, Hydrodynamics, 2nd ed. p. 474. 



Helmholtz's Formula 189 

EXAMPLES 

1. A wave-function u is required to satisfy the following initial conditions for t = 

u=f(x, y), -^ = when 2 = 0, 
u = o, = when z ^ 0. 

C/I 

Prove that u is zero when z 2 > c 2 t 2 and when z 2 < c z t 2 u = / where /denotes the mean value 
of the function / round that circle in the plane 2 = whose points are at a distance ct from 
the point (x, y, 2). 

2. If in Ex. 1 the plane z = is replaced by the sphere r = a, where r 2 = x 2 + y 2 -f z 2 , 
the wave- function w is equal to - / when there is a circle (on the sphere) whose points are all 
at distance ct from (x, y, z) and is otherwise zero. 

2-63. Helmholtz's formula. When a wave-function is a periodic 
function of t, Kirehhoff's formula may be replaced by the simpler formula 
of Helmholtz. 

Putting u = U (x, y, z) e lkct 

the wave -equation gives 

V 2 C7 + k*U = 0. 

Applying Green's theorem to the space bounded by a surface S and a 
small sphere surrounding the point (x ly y ly zj we obtain formula (A) 

477 U (x^y^zj =-*/ (*, y, *) (R~ l e~ lkR ) dS + fl-ic-** dS, 



where R* = (x - xtf + (y - yj* + (z - ^) 2 , 

and the normal is supposed to be drawn out of the space under considera- 
tion. This space can extend to infinity and the theorem still holds provided 
U -^ like Ar~ l e~ lkr as r-> oo, r being the distance of the point (x, y, z) 
from the origin. It is permissible, of course, for U to become zero more 
rapidly than this. 

A solution of the more general equation 

V 2 U + k*U + a> (x, y, z) - 
is obtained by adding the term 



f 1 1 R- l e~ lkR a> (x, y, z) dxdydz 



to the right-hand side of (A) and it is chiefly in this case that we want to 
integrate over all space and obtain a formula in which U (x l9 yi, zj is 
represented by this last integral. 

In the two-dimensional case when u is independent of z, the function 
to be used in place of R- l e~ lkR is derived from the function 



*u= fit -cosh a) da, 
Jo \ c / 



190 Applications of the Integral Theorems of Gauss and Stokes 

already mentioned. Writing u = Ue ikct as before, the elementary potential 
function satisfying V 2 U + k 2 U = is K (ikp), where K (ikp) is defined 
by the equation 

K (ikp) = e- ik Kx**da. 

Jo 

This is a function associated with the Bessel functions. For large values 
of R we have . . , 



while for small values of R 

K Q (iR) + log (B/2) 
is finite. The two-dimensional form of Green's theorem gives 

...... (B) 

where p 2 = (x - a^) 2 + (y - 2/i) 2 , 

rf< is an element of the boundary curve and n denotes a normal drawn into 
the region in which the point (x l9 y^ is situated. 
A solution of the more general equation 

V 2 U + k*U + w (x, y) = 
is likewise obtained by adding the term 



J J K (ikp) w(x, y) dxdy 



to the right-hand side of (B). When k = the corresponding theorem is 
that a solution of the equation 

V 2 t7 + eu (x, y) = 
is given by 2-rrU = - I logpoj (x, y) dxdy. 

2-64. Volterra's method*. Let us consider the two-dimensional wave- 
equation r 3 9 3 

L W s -fc-te*-W~ f(Xl *' Z) 

in which 2 = ct. If the problem is to determine the value of u at an 
arbitrary point (, 77, ) from a knowledge of the values of -u and ite 
derivatives at points of a surface /S, we write 

Z=*s- r = 2/-7 ? , Z = z-f, 

and construct the characteristic cone X 2 + Y 2 = Z 2 with its vertex at the 
point (, 77, ^). We shall denote this point by P and the cone by the 
symbol F. 

* Ada Math. t. xvm, p. 161 (1894); Proc. London Math. Soc. (2), vol. u, p. 327 (1904); 
Lectures at Clark University, p. 38 (1912). 



Volterra's Method 191 

Volterra's method is based on the fact that there is a solution of the 
wave-equation which depends only on the quantity ZjR y where 

R 2 = X 2 + 7 2 . 

This solution, v, may, moreover, be chosen so that it is zero on the charac- 
teristic cone F. The solution may be found by integrating the fundamental 
solution (Z 2 X 2 Y 2 )~* with respect to Z and is cosh- 1 w where 
w = Z/R. Since w = 1 on F it is easily seen that v = on F. 

For this wave-equation the directions of the normal n and the co- 
normal v are connected by the equations 

cos (vx) cos (nx), cos (vy) = cos (ny), cos (vz) cos (nz). 

At points of F the conormal is tangential to the surface and since v is 

^77 
zero on F, ~- is also zero. The function v is infinite, however, when R = 

GV 

and a portion of this line lies in the region bounded by the cone F and 
the surface S. We shall exclude this line from our region of integration 
by means of a cylinder (7, of radius c, whose axis is the line R = 0. We 
now apply the appropriate form of Green's theorem, which is 



to the region outside C and within the realm bounded l- >y S and F. On 
account of the equations satisfied by u and v the forgoing equation 
reduces simply to 

JU- *-*)"-- JJJ* fc - 

On C we have 



dS = 
and since lim (s log e) = 

e ->0 

we have Hm j j^ (u | - v^) - - 2 |* tf , ,, ) &, 

where (^, r\> z) is on A$ f and is in the part of 8 excluded by C. We thus 
obtain the formula 



and the value of u at P may be derived from this formula by differentiating 
with respect to . The result is 



192 Applications of the Integral Theorems of Gauss and Stokes 



EXAMPLE 

Prove that a solution of the equation 



dx* + By* = c 2 W 
is given by the following generalisation of Kirchhoff's formula, 

2nu (x, y,t) = / [c 2 (t - tj* - r 2 ]"* {cos nt - cos nr . c (t - tj/r} u fa, ft , fj dS l 

J <r 



f f 



in which r 2 = (x a^) 2 + (y ft) 2 and the integration extends over the area a cut out on 
a surface 8 in the (x 19 y l9 t 1 ) space by the characteristic cone 

fa - *) 2 + (ft - y) 2 = c 2 & - t)\ 

the time t being chosen so as to satisfy the inequality t < t . 

[V. Voltorra.] 

2-71. Integral equations of electromagnetism. Let us consider a region 
of space in which for some range of values of t the components of the 
field-vectors E and H and their first derivatives are continuous functions 
of x, y, z and t. 

Take a closed surface S in this region and assign a time t to each point 
of S and the enclosed space in accordance with some arbitrary law 

t = f(z, y, z), 

where / is a function with continuous first derivatives. We shall suppose 
that this function gives for the chosen region a value of t lying within the 
assigned range and shall use the symbol T to denote the 'vector with 

4. 3 / 3 / 9 / 

components^, ^. 

Writing Q = H + iE as before we consider the integral 



/ = Jj [Q - i (Q x T)] n dS 



taken over the closed surface S. The suffix n indicates the component along 
an outward -drawn normal of the vector P which is represented by the 
expression within square brackets. Transforming the surface integral into 
a volume integral we use the symbol Div P to denote the complete diver- 
gence when the fact is taken into consideration that P depends upon a 
time t which is itself a function of x, y and z. The symbol div Q, on the 
other hand, is used to denote the partial divergence when the fact that 
Q depends upon x, y and z through its dependence on t is ignored. We 
then have the equation 

/ = 1 1 j div P . dr, 



where div P - div Q + T . R, 

3Q 

and R - -57 i curl Q. 

ot 



Integral Equations of Electromagnetism 193 

Now div Q and R vanish on account of the electromagnetic equations 
and so these equations are expressed by the single equation / = 0. When 
/ is constant T = and the equation 7=0 gives 



which correspond to Gauss's theorem in magneto- and electrostatics. It 
may be recalled that Gauss's theorem is a direct consequence of the inverse 
square law for the radial electric or magnetic field strength due to an 
isolated pole. The contribution of a pole of strength e to an element EdS 
of the second integral is, in fact, eda>/4t7T, where cfa> is the elementary solid 
angle subtended by the surface element dS at this pole. On integrating 
over the surface it is seen that the contribution of the pole to the whole 
integral is e, \e or zero according as the pole lies within the surface, on 
the surface or outside the surface. 

This result is usually extended to the case of a volume distribution of 
electricity by a method of summation and in this case we have the equation 



where p denotes the volume density of electricity. 

Transforming the surface integral into a volume integral we have the 
equation 

J j J (div E - p) dr = 0, 

which gives div E = p. 

Since E = V</> the last equation is equivalent to Poisson's equation 

V V + p = 0, 

in which the factor 4?r is absent because the electromagnetic equations 
have been written in terms of rational units. Our aim is now to find a 
suitable generalisation of this equation. In order to generalise Gauss's 
theorem the natural method would be to start from the field of a moving 
electric pole and to look for some generalisation of the idea of solid angle. 
This method, however, is not easy, so instead we shall allow ourselves to 
be guided by the principle of the conservation of electricity. The integral 

which must be chosen to replace P^T should be of such a nature that 

its different elements are associated with different electric charges when 
each element is different from zero. When the elements are associated 
with a series of different positions of the same group of charges which at 
one instant lie on a surface it may be called degenerate. In this case we 

B 13 



194 Applications of the Integral Theorems of Gauss and Stokes 

can regard these charges as having a zero sum since a surface is of no 
thickness. Now it should be noticed that if we write 

= *-/(*,*/, z), 
the quantity 

M SO , W ( 80 , W _ , ^ 

:/; = 27 + v x %- + v v 5- + v o- = 1 (0 T) 
dt dt ox v cy z dz ' 

vanishes when the particles of electricity move so as to keep 6 = 0, that 
is so as to maintain the relation t = / (x y y, z), and in this case the integral 

dd 



is degenerate. 

We shall try then the following generalisation of Gauss's theorem and 
examine its consequences : 

I=i\\\ P [l-(v.T)]dT. 



Transforming the surface integral into a volume integral we have 
= [div Q - ip + T . (R 4- ipv)] dr y 



and since the function / is arbitrary this equation gives 

div Q = ip, R = - ipv. 
Separating the real and imaginary parts we obtain the equations 

3E 
curl H = -fa + />v, div E = p, 

curlE= - I?, divH=0, 
ct 

which are the fundamental equations of the theory of electrons. The first 
two equations give ~ 



which is analogous to the equation of continuity in hydrodynamics. Our 
hypothesis is compatible, then, with the principle of the conservation of 
electricity. The integral equation 

[Q- (Q x T)] n dS= }J|p[l - (v.T)]dr (A) 

will be regarded as more fundamental than the differential equations of 
the theory of electrons if the volume integral is interpreted as the total 
charge associated with the volume and is replaced by a summation when 
.the charges are discrete. This fundamental equation may be used to obtain 
the boundary conditions to be satisfied at a moving surface of discontinuity 
which does not carry electric charges. 

Let t = f (x, y, z) be the equation of the moving surface and let the 



Boundary Conditions 195 

surface 8 be a thin biscuit-shaped surface surrounding a superficial cap S 
at points of which t is assigned according to the law t =/(#, y, z). At 
points of the surface S we shall suppose t to be assigned by a slightly 
different law t = / x (#, t/, z) which is chosen in such a way that the points 
of S on one face have just not been reached by the moving surface 
t = f (x, y, z), while the points on the other face have just been passed over 
by this surface. Taking the areas of these faces to be small and the thick- 
ness of the biscuit quite negligible the equation (A) gives 

[Q' - (Q' x T)] = [Q" - i (Q" x T)] n , 

where Q', Q" are the values of Q on the two sides of the surface of dis- 
continuity and the difference between / x and / has been ignored. Writing 
q = Q' Q" we have the equation 

[q - * (q x T)] n = 0. 
Now the direction of the cap $ is arbitrary and so q must satisfy the 

relation Q-.'(qxT). 

This gives q 2 = 0. Hence if q = h + ie we have the relations 

h* - & = 0, (h.e) = 0. 
The equation also gives (q . T) = 0, 

and (q x T) = i (q x T) x T - i [T (q . T) - qT 2 ] 

= - iT 2 q or T 2 = 1 if q ^ 0. 

Hence the moving surface travels with the velocity of light. 

A similar method may be used to find the boundary conditions at the 
surface of separation between two different media. We shall suppose that 
the media are dielectrics whose physical properties are in each case specified 
by a dielectric constant K and a magnetic permeability /*. For such a 
medium Maxwell's equations are 

, divD-0, 



^- 

where D = #E, B 

Instead of these equations we may adopt the more fundamental 
integral equations 



J| 



[B + (E : 

which give the generalisations of Gauss's theorem. The boundary con- 
ditions derived from these equations by the foregoing method are 

d - (h x T) = 0, b + (e x T) = 0, 

13-2 



196 Applications of the Integral Theorems of Gauss and Stokes 

where e, h, d, b are the differences between the two values of the vectors 
E, H, D, B respectively on the two sides of the moving surface. These 
equations give (A . T) = , (b . T )^0, 

(d x, T) + T*h = (h . T)T; (b x T) - T 2 e - - (e . T) T. 
If the vector v represents the velocity along the normal of the moving 
surface we have v ^ ^ y V T = \ 

hence the equations may be written in the form 

d v = 0, b v = 0, h r = (v y d) r , e T = (v x b) T , 

where d v , b v denote components of d and 6 normal to the moving surface 
and the suffix r is used to denote a component in any direction tangential 
to the moving surface. When this surface is stationary the conditions take 
the simple form 

d v = 0, b = 0, h r = 0, e r = 

used by Maxwell, Rayleigb and Lorentz. 

When a surface of discontinuity moves in a medium with the physical 
constants K and /z, we have Heaviside's equations (Electrical Papers, 
vol. ir, p. 405) X(e.T) = 0, /*<h.T) = 0, 

K (e x T) + T 2 h - 0, p. (h x T) - T 2 e = 0,. 
and so A> [(h x T) x T] = KT* (e x T) - - T 4 h, 

i.e. Kfji - T 2 

if h ^ 0. 

The surface thus moves with a velocity v given by the equation 



2-72. The retarded potentials of electromagnetic theory. The electron 

equations , ~ 

i [ (j ty \ 

curl H = - { ~ + pv), div E = p, 

C \ ut I 

curl E = - -37, div // = 

c 3 

may be satisfied by writing 

c o 
where the potentials A and O satisfy the relations 



Retarded Potentials 197 

The last equations are of the type to which Kirchhoff 's formula is applicable 
and so we may write 

...... (B) 



These are the retarded potentials of L. Lorenz. 

The corresponding potentials for a moving electric pole were obtained 
by Lienard and Wiechert. They are similar to the above potentials except 
that the quantity c/M of 1-93 takes the place of 1/r. Let (), ^ (t), 
(t) be the co-ordinates of the electric pole at time t and let a time r be 
associated with the space-time point (x, y, z, t) by means of the relations 

[X - t (T)] +[y-r, (T)] + [z - (T)] = c* (t - T), r < t, (C) 
then 

M=[x-t (r)} ? (r) + [y-r, (r)]r,' (r) + [z - I (r)] ' (r) - c' ( - r), 
and if e is the electric charge associated with the pole the expressions for 
the potentials are respectively 

A --* >( ?\ A -_?9l< T ) A -_< *-_ e - C 
x ~ '' ~ ' z ~ ' 



These satisfy the relation (A) and give the formulae of Hargreaves 

,-, e d (or, T) _ e 9 (a, r) 

x = 4^c 3 (x, ' * = ITT 3ly, z) ' 
where 

o = [* - f (r)] f' (r) + [y - i, (T)] ," (T) + [z - (T)] C" (T) 

+ C* - [' ( T )] - [,' (T)] - [' (T)]. 

It should be remarked that the retarded potentials (B) can be derived 
from the Lienard potentials by a process of integration analogous to that 
by which the potential function 



== ttl 



is derived from the potential of an electric pole. 

Instead of considering each electric pole within a small element of 
volume at its own retarded time r we wish to consider all these electric 
poles at the same retarded time T O belonging, say, to some particular pole 
(&> >?o> o, TO)- Writing 

f W = > (a) + a (a), rj (a) = rj Q (a) + ft (a), (or) = (a) + y (a), 
where a (a), ft (a), y (a) are small quantities, we find that if T is defined by 
(C), 
(r - T ) [(x - &) & +(y- % ) V + (2 - Co) &,' - c ( - T )] 

+ (*-&)+(- %) ^ + (z - ) 7 = 0, 

where , ij , , ^ '> ^o'> o'> a > /S> y are all calculated in this equation at 
time T O . 



198 Applications of the Integral Theorems of Gauss and Stokes 

On account of the motion the pole (, 77, ) occupies at time r the 
position given by the co-ordinates 

= fo + + (r - T ) &', 

*? = i?o + j8 + (T - T ) V. >a 

= o + y + (r - T ) '. 

- 1 ^'* 



If p is the density of electricity when each particle in an element of volume 
is considered at the associated time T and p is the density when each 
particle is considered at time T O , we have 

pd (f, T?, ) - Po d (a, j8, y). 

C7" 

Therefore o == p -- , 

ru r cr (r . v) 

and so f I f ^ dxdydz - f f f -^- , dxdydz. 

c JJJ ^ JJJ^-(^.V) 

Writing p dxdydz ^ de we obtain the Lienard potentials. 

Similar analysis may be used to find the field of a dipole which moves 
in an arbitrary manner with a velocity less than c and at the same time 
changes its moment both in magnitude and direction. 

Let us consider two electric poles which move along the two neigh- 
bouring curves 

* = (*), y-^W, * = (0> 

x = f(t)+a (t), y = TJ (*) + cjB (t), z = J (0 + ey (0, 
e being a quantity whose square may be neglected. If T X is defined in terms 
of x y y, Zy t by the equation 



and T! = T + 0, we easily find that 

JM + a ( T ) [ x - ( T )] + ft ( r ) [y _ , ( T )] + y ( T ) [ z _ J ( T )] = Q. 

If 3/ x is the quantity corresponding to M , we have 

J^ = Jf + c [^^CT + (x - f)a' + (y - T?) $ -f (^ - t) / - f - fa' - y '] 

= Jlf -f [&Jf<7 + p], 

say, where a has the same meaning as before and primes denote differentia- 
tions with respect to T. Now if 

_ eff'frJ + ^Kll ,.,,_ ec 

' ~ 



-_ 

477J/' 



Moving Electric and Magnetic Dipoles 199 

we have 

a x = [A.' -A x }=- , [Ma' - p? + MB?' - MOaf], 



But Ma! - p? + M6(" - M6a' s (y - r,) n' - (z - Q m' - c 2 (t - T) ' 

+ c 2 - nrf + m' + o{a(t- T) n(y ij) + m(z )} 
where I = ft' - m', m =* y? - oj', = <nj' - #'. 

Hence we may write 

_ e riL^ 7 ^ _ i f!M 0.1 f-^i 

a * ~ ~ 47r [fy \M) dz \MJ + 9< \M)\ ' 

ec p /\ a //?\ a/ y \i 
^ ~ 4^ [ai UJ + % \M) + dz \M)\ 

These results may be obtained also with the aid of the general theorem 
which gives the effect of an operation -r analogous to differentiation, 

i r i d n = i n ^ 



i d n = i n ^DI , i n i 

f drj ao; [M d (e, r) J " dy lMd(,r 



/ being a function of r and 6. Writing / = f , ^- = a, =-' = /J, ^- = y the 

expression for a^ is at once obtained from that for ^4 X . Writing/ = r we 
obtain the expression for (f>. 

The formulae show that the field of the moving dipole may be derived 
from Hertzian vectors II and F by means of the formulae 



where u and w are vector functions of r with components (a, j3, y), (Z, m, n) 
respectively. If v denotes the vector with components (', T\ ', x ) we have 
the relations ( v . w) = 0, (u . W) - 0, 

consequently Hertzian vectors of types (D) do not specify the field of a 
moving electric dipole unless these relations are satisfied. 

Since A = - 37 + curl T, B = - ^ - curl II, 

c ot c ot 



where B and O are the electromagnetic potentials of magnetic type, we 
may write down the potentials for a moving magnetic dipole by analogy. 



200 Applications of the Integral Theorems of Gauss and Stokes 

We simply replace II by T and F by II. Hence the potentials of a moving 
magnetic dipole are of type 



a - - -- 

x ~ 4n dy \M) dz \M) dt \M 

me f" 3 / / \ 9 fm\ d / n 
~ 



Let us now calculate the rate of radiation from a stationary electric 
dipole whose moment varies periodically. Taking the origin at the centre 
of the dipole, we write 

HX = ^/(T), H v =g(T) 9 il z = - r h(r), r=-r/c, 

where /, y, h are periodic functions with period T. The vector is zero since 
there is no velocity. 

In calculating the radiation we need only retain terms of order 1/r in 
the expressions for E and H . To this order of approximation we have 



=- (yE, - zE y ), 



where ^ = ^ (/"* + g"* + h"*) - ~ (xf" + yg" + zh")*. 

The rate of radiation is obtained by integrating cE 2 over a spherical 
surface r = a, where a is very large. With a suitable choice of the axis 
of z we may write 



and the value of the integral over the sphere is 



the mean value over a period T is 

A 2 /27T\ 4 

" " 



Electromagnetic Radiation 201 

EXAMPLE 

and L (x, y, z, t, s) = [x - (s)] I (s) + [y ^ (5)] m (*) -f [2 - (*)] n (s) - c(t ~ s), 
prove that the potentials 

1 i T ft i \ l(s)ds v'( T ) 

* = 4w J - oo (#, y, 2, /,~) " ' ( 4^ ' 

4 _ 1 f r ///^ m W^ 



/ ^' (*,*,,*,*,*) ' V ''4^' 

are wave-functions satisfying the condition 

1V "*" c dt = * 

Show also that in the field derived from these potentials the charge associated with the 
moving point (T), y (T), (T) is/(r), the variation with the time being caused by the 
radiation of electric charges from the moving singularity in a varying direction specified by 
the direction cosines / (r), m (T), n (T). 

2-73. The reciprocal theorem of wireless telegraphy. If we multiply the 
electromagnetic equations 

JSj = curl (cEi), C 1 = curl (c^) 

for a field (E ly H^ by H 2 , E 2 respectively, where (E 2 , H 2 ) are the field 
vectors of a second field in the same medium, and multiply the field 

equations A , , ^ x ~ i / r? x 

^ B 2 = curl (cE 2 ), C 2 = curl (cH 2 ) 

for this second field by H 19 E l respectively and then add all our equations 

together, we obtain an equation which may be written in the form 

(H 2 . B,) - (H, . B 2 ) + (E l . C 2 ) - (E 2 . C,) = cdiv (E 2 x HJ - cdiv (E I x H 2 ). 

(A) 

We now assume that both fields are periodic and have the same frequency 
co/277. Introducing the symbol T for the time factor e~ ltat and assuming 
that it is understood that only the real part of any complex expression 
in an equation is retained, we may write 

TT WL u nnjt ~& ^n^t & nn^i 
/i-i==j[/ij,. juL 2 j. n 2 j /vj = JL 6j , jG/2 ,JL e 2) 

where the vectors e ly h l9 c l9 etc. depend only on x, y and z. 

Now let K, fji and a be the specific inductive capacity, permeability and 
conductivity of the medium at the point (x, y, z) and let a denote the 
quantity (ioo -f icr)/c, then we have the equations 

B 2 == ia)jjih 2 T, C l = iceiTa, C 2 = ice 2 Ta, 



202 Applications of the Integral Theorems of Gauss and Stokes 

which indicate that the left-hand side of equation (A) vanishes. The 
equation thus reduces to the simple form * 

div (e 2 x hj) = div (e 1 x h 2 ). 

This equation may be supposed to hold for the whole of the medium 
surrounding two antennae f if these sources of radiation are excluded by 
small spheres K l and K 2 . An application of Green's theorem then gives 

[ (% x ^i)n dS -f [ (e 2 x hi) n dS - I (e l x h 2 ) n dS + [ (e l x h 2 ) n dS, 

J KI J K t J KL J KZ 

an equation which may be written briefly in the form 

J n -f- J 21 J 12 4- </22- 

Let the first antenna be at the origin of co-ordinates and let us suppose 
for simplicity that it is an electrical antenna whose radiation may be 
represented approximately in the immediate neighbourhood of O by the 
field derived from a Hertzian vector TIT with a single component H Z T, 
where 



= Me lpR (c 2 p 2 = kfjLa> 2 -f 

and M is the moment of the dipole. In making this assumption we assume 
that the primary action of the source preponderates over the secondary 
actions arising from waves reflected or diffracted by the homogeneities of 
the surrounding medium. 

Using % to denote the value of a at and writing FT for 3FI/9r, ( 2 > y* > 2) 
for the components of e 2 , we have 



n = - ia, f {(** 

J K, 



-f y 2 ) 2 - a*& - yz^} H'R~*dS. 



For the integration over the surface K l the quantities 7? 2 , Il x and the 
vector e 2 may be treated as constants, for K 1 is very small and e 2 varies 
continuously in the neighbourhood of 0. We also have 



[yzdS - \zxdS = \xydS = 0, (x*dS= ^dS= \z 2 dS = 
IxdS^O, lx*dS = 0, l 



Therefore J n = - 



and, since lim 

.R->0 

we have finally J u = 2ia 1 M 1 2 /3. 

Now the rate at which the antenna at radiates energy is 
8 = ^W 2 127rF 3 , V = c (x)'i. 



* H. A. Lorentz, Amsterdam. Akad. vol. iv, p. 176 (1895-6). 

f A. Sommerfeld, Jahrb. d. drahtl Telegraphie, Bd. xxvi, S. 93 (1925); W. Schottky, ibid. 
Bd. xxvn, S. 131 (1926). 



Reciprocal Relations 203 

Assuming that 8 is the same for both antennae we obtain the useful 
expression 



The integral J 21 is seen to be zero because it involves only terms which 
change sign when the signs of x, y and z are changed. 

Evaluating / 22 and J u in a similar way we obtain the equation 

(^ + icrjaj) (VtfK^ 2 = (ic, + iaju) (F 2 3 //c 2 )i &, 

where f 1? rj l9 x are the components, at the second antenna 2 , of the vector 
e lf The amplitudes and phases of the field strength received at the two 
antennae are thus the same when both antennae are of the electric type 
and are situated at places where the medium has the same properties and 
emits energy at the same rate. When the two antennae are both of magnetic 
type the corresponding relation is 

(M 2 F 2 3 )*ri=(Mi^ 3 )*y 2 , 

where a^ , & , y^ are the components of h t and a 2 , /J 2 , y 2 are the components 
of A 2 . The antennae are again supposed to be directed along the axis of 
z but there is a more general theorem in which the two antennae have 
arbitrary directions. 

The relation (A) and the associated reciprocal relation remind one 
of the very general extension of Green's theorem which was given by 
Volterra* for the case of a set of partial differential equations associated 
with a variational principle. This extension of Green's theorem is closely 
connected with a property of self-adjointness which has been shown by 
Hirsch, Kiirsch&k, Davis and La Pazj to be characteristic of certain equa- 
tions associated with a variational principle. In the case of the Eulerian 
equation F = associated with a variational principle 87 = 0, where 

l>i,3 2 , Xn\u\ u lt u 2j ... u n ]dx l dx 2 ...dx n , 

du d*u . 1 

Us = 7^ > u " = ^T^r> ( r > s = !> 2 > n )> 

x UJsg , CJC r VJUg 

the equation which is self -adjoint is the "equation of variation" for v 

dF 3F dF dF 3F 

= v = -- h ^ -A -- h ... v n ~ -- h v n -- --- h v 12 x- - + ... . 

du 1 dut du n du n li du 12 

* V. Volterra, Rend. Lincei (4), t. vi, p. 43 (1890). 

f A. Hirsch, Math. Ann. Bd. XLIX, S. 49 (1897); J. Kurschdk, ibid. Bd. LX, S. 157 (1905); 
D. R. Davis, Trans. Amer. Math. Soc. vol. xxx, p. 710 (1928); L. La Paz, ibid. vol. xxxir, p. 509 
(1930). 



CHAPTER ^ 
TWO-DIMENSIONAL PROBLEMS 

3-11. Simple solutions and methods of generalisation of solutions. A 
simple solution of a linear partial differential equation of the homo- 
geneous type is one which can be expressed in the form of a product of a 
number of functions each of which has one of the independent variables 
as its argument. Thus Laplace's equation 

927 



== 

dx 2 dy 2 
possesses a simple solution of type 

V = e~ my cos m(x - x'), ...... ( A ) 

where m and x' are arbitrary constants ; the equation 

9F_ d 2 V 
~ft- K ~dx* 

ppssesses the simple solution 

V = e~ Km2t cos m (x - x'}, ...... (B) 

and the wave-equation 



^ v 
dx 2 ^ c 2 



possesses the simple solution 



V = sin met cos m (x x'). ...... (C) * 

itv 

The last one is of great historical interest because it was used by Brook 
Taylor in a discussion of the transverse vibrations of a fine string. It 
should be noticed that the end conditions F = when x = a/2 are 
satisfied by a solution of this type only if ma = 2n -f 1, where n is an 
integer. There are thus periodic solutions of period 

T = 27r/wc - 2ira/(2n -f 1). 

If M (m., x') denotes one of these simple solutions a more general 
solution may be obtained by multiplying by an arbitrary function of m 
and x' and then summing or integrating with respect to the parameters 
m and x'. This method of superposition is legitimate because the partial 
differential equations are linear. When infinite series and infinite ranges 
of integration are used it is not quite evident that the resulting expression 
will be a solution of the appropriate partial differential equation and some 



Generalisation of Simple Solutions 205 

process of verification is necessary. If, for instance, we take as our generalisa- 
tion the integral 

V = f M (m, x')f(m] dm (t > 0, y > 0), 

Jo 

and distinguish between solutions of the different equations by writing 
v for V when we are dealing with a solution of the second equation and 
y for V when we are dealing with a solution of the third equation, we easily 
find that when / (m) = 1 we have 



2v = (77//c)*exp [ (x x')*lKt], 
y = - , - or according as | x x' \ = c ( > 0). 

It is easily verified that these expressions are indeed solutions of their 
respective equations. These solutions are of fundamental importance 
because each one has a simple type of point of discontinuity. In the last 
case the points of discontinuity for y move with constant velocity c. 

We may generalise each of these particular solutions by writing V, v 
or y equal to 

M (m, x') F (x f ) dx'dm, 

Jo J -oo 

where the integration with regard to x' precedes that with respect to m. 
When the order of integration can be changed without altering the value 
of the repeated integral the resulting expressions are respectively 

F= p yF(x')dx' 



2v = (77/Ac*)* I"" exp [- (x - x')*lK(\ F (x f ) dx', 

J -00 

fjr + ct 

y=l\ F(x')dx'. 

* Jx-ct 

The last expression evidently satisfies the differential equation when F (x) 
is a function with a continuous derivative; y represents, moreover, a 
solution which satisfies the conditions 



when t = 0. 

When the function F (x) is of a suitable type the functions V and v also 
satisfy simple boundary conditions. This may be seen by writing 

x' - x + y tan (6/2) 
in the first integral and x ' =* x + 2u (irf)* 



206 Two-dimensional Problems 

in the second. The resulting classical formulae 



Too 

v = TT* F[x+2u (AC 

J -00 



u 



suggest that V = nF (x) when t/ = and v = -nF (x) when 2=0. 

These results are certainly true when the function F (x) is continuous 
and integrable over the infinite range but require careful proof. The 
theorems suggest that in many cases * 

rrF (x) = [ dm f cos m(x- x') F (x') dx'. 

Jo J -oo 

This is a relation of very great importance which is known as Fourier's 
integral theorem. Much work has been done to determine the conditions 
under which the theorem is valid. 

A useful equivalent formula is 



(x) = I dm f e tm( *-*'> F (x f ) dx'. 

J oo J oo 



When F (x) is an even function of x Fourier's integral theorem may be 
replaced by the reciprocal formulae 



f 00 

F (x) cos mxG (m) dm, 

Jo 

2 f 

(m) =- co$mxF(x)dx, 

7T Jo 



and when F (x) is an odd function of x the theorem may be replaced by 
the reciprocal formulae 

f 

F (x) = sin mxH (m) dm, 
Jo 

2 f 
H (m) = - sin mxF (x) dx. 

TT JO 

The formulae require modification at a point x, where F (x) is discon- 
tinuous. It F (x) approaches different finite values from different sides of 

* The theorem is usually established for a continuous function which is of bounded variation 

fee A) 

and is such that / | F (x) \ dx and I | F (x) \ dx exist. F (x) may also have a finite number of 

/ -00 J -JO 

points of discontinuity at which F (x + 0) and F (x - 0) exist but in this case the integral represents 

' J [[F(x+ 0)+ F (x - 0)]. Proofs of the theorem are given in Carslaw's Fourier Series and 

2 

Integrals; in Whittaker and Watson's Modern Analysis; and in Hobson's Functions of a Eeal 

Variable. 



Fourier' '$ Inversion Formulae 207 

the point x the integral is found to be equal to the mean of these values 
instead of one of them. Thus in the last pair of formulae we can have 

F(x)=l xf>, H(m)= 2 -[l-cosw], 

77 
= X> (f>, 

but the integral gives F (1) = \ . 



EXAMPLE 

If S (x, t) = (7r/c<)~i exp [- x*/4:Kt] and / (x) is continuous bit by bit a solution of 
%r- = K x- 2 , which satisfies the condition y = f (x) when < = and oo < jc < oo , is 
given by the formula 

v = 4 T flf (x - * , 0/(s ) ^o + 4 3 [/ (* n ) - f(xn)] 

J oo n=l 

x /J [S (a? -a; n -,)- fl (* - s,, + , *)] <fc 

where 2/ (#) = / (z n + 0) + / (x n 0) and the summation extends over all the points of 
discontinuity of/ (x). 

3-12. A study of Fourier's inversion formula. The first step is to 
establish the Biemann-Lebesgue lemmas*. 

Let g (x) be integrable in the Riemann sense in the interval a < x < b 
and when the integral is improper let | g (x) \ be integrable. We shall 
prove that in these circumstances 

f b 

lim sin (kx) . g (x) dx = 0. 

k ->w J a 

Let us first consider the case when g (x) is bounded in the range (a, 6) 
and G is the upper bound of \g (x)\. We divide the range (a, 6) into n 
parts by the points # 15 x 2 , ... x n _! and form the sums 

S n = U, (x, - a) + U 2 (x 2 - x,) + ... U n (b - x n ^) 9 

s n L (Xi ~ a) 4- L 2 (x 2 x^ -f ... L n (b x n _j), 

where C7 r , L r are the bounds of g (x) in the interval se rHl < x < x r , so that 
in this interval 

Since & (x) is integrable we may choose n so large that S n s n < c, where 
c is any small positive quantity given in advance. Now 

g (x) sin Icxdx = S g r (x r ^) sin kx . dx + 2 oj r (x) sin kx ' dx 

* The proof in the text is due to Prof. G. H. Hardy and is based upon that in Whittaker and 
Watson's Modern Analysis. 



208 



Two-dimensional Problems 



the summations on the right being from r = 1 to r = n and the integrations 
from # r _, to x r . With the same convention 

r b 



I ( b 

g (x) sin kxdx 

I Ja 



sin kx . dx 



+ 



< 2*167* -f S n - s n < 2nG/k -f c. 

Keeping ?i fixed after e has been chosen and making k sufficiently large 
we can make the last expression less than 2e and so the theorem follows 
for the case in which g (x) is bounded in (a, b). When g (x) is unbounded 
and | g (x) \ Integra ble in (a, b) we may, by the definition of the improper 
integral, enclose the points at which g (x) is unbounded in a finite number 
of intervals i l9 i 2 , ... i p such that 

P 

S I g (x) I dx < . 

r~lJi, 

Now let G denote the upper bound of g (x) for values of x outside the 
intervals i r and let e 1? <? 2 , ... e p+l denote the portions of the interval (a, b) 
which do not belong to i lt i 2 > ... i v , then we may prove as before that 



g (x) sin kx . dx 



g (x) sin kx . dx 



P r I 

-f- 2 \ g (x) sin kx 



dx 



< 2nG/k 4- 2e. 

Now the choice of e fixes n and 6r, consequently the last expression 
may be made less than 3e by taking a sufficiently large value of k. Hence 
the result follows also when g (x) is unbounded, but subject to the above 
restriction. 

Some restriction of this type is necessary because in the case* when g (x) 
is the unbounded function x~ l (1 x 2 )"* for which | g (x) \ is notintegrable 
in the range ( 1, 1) we have 

j-l fA: 

sin kx g (x) dx '= TT J (r) rfr, 
J - 1 Jo 

r A r co 

and as k -> oo J (r) dr = J Q (r) dr = 1. 

Jo Jo 

The next step is to show that if x is an internal point of the interval 
(a, /?), where a and /J are positive, and if / (x) satisfies in (a, j8) the 
following conditions : 

(1) f (x) is continuous except at a finite number of points of dis- 
continuity, and if / (x) has an improper integral | / (x) \ is integrable ; 

(2) / (x) is of bounded variation, then 



limf 






1C -^oo J - a 

Let us write 






sin k (t x) 



[x - 0)] = ^ 

- L +i: 



say. 



Fourier's Integrals 209 

and transform the integrals by the substitutions t = x u and t = x -f- u 
respectively, then 

r/ 3 



r 55* ( 

J a & 

fa 

Jo 



-f- 



f J? ai 
^ 

Jo 



fa-t-jr 

_^)_/(z_0)]c^4-/(z- 0) sin ku. du/u 

Jo 

r&-x 

+ ^) -/(* + Q)]du+f(x+ 0) sinfai.dWtt 

Jo 



Now let c denote one of the two positive quantities a -f x, j3 x, then 

fc r l:c n 

sin ku . du/u = sin v . dv/v -> ^ as fc -> oo. 
Jo Jo * 

Also, let F (u) denote one of the two functions / (x u) - / (x 0), 
/ (x + w) / (x -j- 0), then jP (0) = and .F (u) is of bounded variation in 
the interval (0 < u < c). We may therefore write 

F (u) = HI (u) - // 2 (u), 
where H l (u) and H 2 (u) are positive increasing functions such that 

H, (0) = a, (0) = o. 

Given any small positive quantity 6 we can now choose a positive 
number z such that 

< H l (u) < e, < # 2 (u) < e, 

% 

whenever < u < z. We next write 

sin ku . F (u) du/u = sin ku . F (u) du/u 

Jo J z 

-f sin ku . H \ (u) du/u sin ku . H 2 (u) du/u. 

Jo Jo 

Let H (ut) denote either of the two functions H l (u), H 2 (u)] since this 
function is a positive increasing function the second mean value theorem 
for integrals may be applied and this tells us that there is a number v 
between and z for which 

sin ku . H (u) du/u = H (z) \ sin ku . du/u 

Jo Jv 

I ( kz 

H (z) sin s . dsls . 

I Jkv 

roo r oo 

Since sin s . dsjs is a convergent integral, sin s . dsjs has an upper 

JO JT 

bound B which is independent of r and it is then clear that 



11; 



sin ku . H (u) du/u 



< 2BH (z) < 2B . 



210 Two-dimensional Problems 

By the first lemma k may be chosen so large that 

sin ku . F (u) du/u < e, 
and so we have the result that 

re 

lim sin ku . F (u) du/u = 0. 

Ar-voo Jo 

It now follows that 

k -><X> J -a * X 4 

To extend this result to the case in which the limits are oo and oo 
we shall assume that for x > j3 



where P 1 (x) and P 2 (x) are positive functions which decrease steadily to 
zero as x increases to oo. A similar supposition will be made for the range 
x < a, the positive functions now being such that they decrease steadily 
to zero as x decreases to oo. Since 



Pi(t) 
t- x' 



(x < /? < t < y) 



is a positive decreasing function of t for t> ft we may apply the second 
mean value theorem for integrals and this tells us that 

x) ^p l( p) {* sink(t _ x)dt + P> 

- 



Now let \P l (x)\< M for x > /?, then 

M 



D m ,, 

P l (t)dt 



J-TQ - , 
k(p-x) 



-f 



sin 



By making k large enough we can make 4M/k (j8 x) as small as we 
please ; moreover, this quantity is independent of y, and so we can conclude 

that ("sink (t-x) 

lim I _ P x (t) dt 0. 

k ->oo J P t 

Similar reasoning may be applied to the integral involving P 2 (t) and 
to the integrals arising from the range t < a. It finally follows that 



foo 

im 

_>00 J-C 



lim 



or 



sin k (t x) 

x, t x 

}oo rk 
dt ' 
-oo JO 



~ x)f(t)ds. 



Fourier's Integrals 211 

To justify a change in the order of integration it will be sufficient to 
justify the change in the order of integration in the repeated integral 



I dt f cos s (t - x) P l (t) ds, 

Jq JO 



where q > /?, for the other integral with limit oo may be treated in the 
same way and a change in the order of integration for the remaining 
integral between finite limits may be justified by the standard analysis. 
Now let us assume that 

\ m Pi(t)dt ...... (A) 

Jq 

exists, then 
f "dt j k cos s(t- x) P l (t) ds~ \ K ds Tcos s(t-x) P l (t) dt < 2k f "X (t) dt. 

Jq JO JO Jq Jq 

But, since the integral (A) exists we can choose q so large that 



is as small as we please. The order of integration can therefore be changed 
and so we have finally 



TT/(X) = \ds\ coss(t~ x)f(t) dt. 

JO J -oo 



The assumptions which have been made are: 

(1) For x > p, f(x) = Pj (x) - P 2 (x), where P 1 (x) and P 2 (x) are 
positive decreasing functions integrable in the range ()8, oo). 

(2) A corresponding supposition for x < a. 

(3) / (x) of bounded variation in a range enclosing the point x. 

(4) / (x) discontinuous at only a finite number of points in ( a, f3) 
and j / (x) | integrable in ( a, /2). 

3-13. To illustrate the method of summation we shall try to find a 
potential which is zero when x = and when x = 1 . We shall be interested 
here in the case when the potential has a logarithmic singularity at the 
point x = x', y = 0. 

We first note that M (m, x') is a simple combination of primary 
solutions and by an extension of the method of images used in the solution 
of physical problems by means of primary solutions we may satisfy the 
boundary condition at x = by means of a simple potential of type 
M (m, x') M (m, x'). This can be written in the form 

2e~ my sin (mx) . sin (mx')> 

and it is readily seen that the boundary condition at x = 1 may be satisfied 
by writing m = mr, where n is an integer. We now multiply by a function 

14-2 



212 Two-dimensional Problems 

of n and sum over integral values of n. To obtain a series which can be 
summed by means of logarithms we choose/ (n) = l/n so that our series is 

co 1 

K = - e~ nny [cos /ITT (x x') cos n-n (x -f #')] 
n~-i n 

If // > the sum of this scries is* 

p i , cosh (rry) cos TT (# 4- a; 7 ) 
~ COsh (TT?/) COS TT (x ~ x') ' 

To extend our solution to negative values of y we write it in the form 

on O 

V g~w;r|y| s j n { n7lX } sin 



The expression for V may be written in an alternative form 



which shows that it may be derived from two infinite sets of line charges 
arranged at regular intervals. 

This expression shows also that the potential V becomes infinite like 
\ log [(x x') 2 -f i/' 2 ] in the neighbourhood of x = x' ', y 0, it thus 
possesses the type of singularity characteristic of a Green's function and 
so we may adopt the following expression for the Green's function for the 
region between the lines x 0, x 1, when the function is to be zero on 

these lines oo j 

G (x,x'\y,y') = - e - nrr \v-v'\ s [ n ( nnx ) sin 

n - 1 n 

A corresponding solution of the equation 



is obtained by writing 

exp [ | y y' \ (nV 2 
in place of exp [ UTT \ y y' \] 

and 2n/(n* - k*/7r 2 ) 

in place of the factor 2/n. 

3-14. As another illustration of the use of the simple solutions of 
Laplace's equation we shall consider the problem of the cooling of the fins 
of an air-cooled airplane engine when the fins are of the longitudinal type. 

The problem will be treated for simplicity as two-dimensional. 

A fin will be regarded as rectangular in section, of thickness 2r, and of 
length a. Assuming that the end x = is maintained at temperature by 
the cylinder of the engine and that it is sufficient to assume a steady state, 

3 2 9 o 2 6 
the problem is to find a solution of ^~ 2 -f ^ = and the boundary con- 

* See, for instance, T. Boggio, Rend, ' Lombardo (2) 42:611-624 (1909). 



Cooling of Fins 213 

ditions * -- = along y = 0, k ~ = q0 along y = T, A y- = <?# along 
# = a. 

The first two of these three conditions are satisfied by writing 

00 

0=2 A m cosh [s m (x-c m )] cos (s m y), ks m r tan (s m r) - q. 

m-l 

This equation gives oo 1 values of s m and when s m has been chosen the 
corresponding value of c m is given uniquely by the equation 

ks m tanh [s m (c m - a)] = y 
which will ensure that the third condition is satisfied. 

To make = Q when x = we have finally to determine the constant 
coefficients A m in such a way that 

00 

= S ^4 W cosh (s m c m ) cos (s m ?/). 
/ - 1 

This may be done with th aid of the orthogonal relations 
I cos (ys m ) cos (ys n ) dy = 0, m ^ n 



m = n. 



Therefore A m = 40 sech 



cosh (^y) sin (s m r). 

v inj) Vm ; 



Harper and Brown derive from this expression a formula for the 
effectiveness of the fin, which they define as the ratio H/H Q) where 

H n = 2q (a + r) n , H = a \0dS. 



For numerical computations it is convenient to adopt an approximate 
method in which the variation of 9 in the y direction is not taken into 
consideration. Results can then be obtained for a tapered fin. 

The approximate method has been used by Binnie| in his discussion 
of the problem for the fins of annular shape which run round a cylinder 
barrel. 

3-15. For some purposes it is useful to consider simple solutions of 
a complex type. Thus the equation 

dv _ d 2 v 
Si = " dx* 

* The formal solution is obtained by D. R. Harper and W. B. Brown (N.A.C.A. Report, 
No. 158, Washington, 1923), but is not used in their computations, 
f Phil. Mag. (7), vol. n, p. 449 (1926). 



214 Two-dimensional Problems 

is satisfied by v = Ae"** (l+l >**, 

if 2i//? 2 = cr. Retaining only the real part we have* 

v = Ae-* x co& (at - px). ...... (A) 

This solution is readily interpreted by considering a viscous liquid which 
is set in motion by the periodic motion of the plane x = 0, the quantity 
v being velocity in one direction parallel to this plane ( 2-56). The pre- 
scribed motion of the plane x = is 

v = A cos at = F, say. 

The vibrations are propagated with velocity or/j3 in the direction per- 
pendicular to the plane but are rapidly damped , for the amplitude diminishes 
in the ratio' e~ 2rr as the wave travels a distance of one wave-length 27r/j3. 
For an assigned value of a this wave-length is very small when v is very 
small, when v is assigned the wave-length is very small if a is very large. 

The equation (A) has b,een used by G. I. Taylorf-to represent the range 
of potential temperature at a height x in the atmosphere, the potential 
temperature being defined as usual, as the temperature which a mass of 
air would have if it were brought isentropically (i.e. without gain or loss 
of heat and in a reversible manner) to a standard pressure. 

The following examples to illustrate the use of the solution (A) are 
given by G. Greenf. 

Suppose that two different media are in contact, the boundary surface 
being x -= a and the boundary conditions 

v 9^, 3 V 2 x 

v > = "" ** = *& for *= a - 

Let there be a periodic source of "plane -waves" on the side x, then 
the solution is of type 

Vl = 6e~^ x cos (at - fax) -f AOeW*-**) cos [at + fa(x 2a)], x < a, 

v 2 - Bde-t* ( *- c) cos [at - fa (x - c)], x > a, 
where c = a[l - (i/^)*], fa = (cr/2^)*, & = (<7/2i/ 8 )*, 

pA = K, vS - #2 vX> PB = 2^ yV 2 > P = #1 V" 2 + #2 vX- 
There is, of course, the physical difficulty that the expression for the 
incident waves becomes infinite when x = oo. 

If we take the associated problem in which the incident waves corre- 
spond to a periodic supply of heat q cos at at the origin, the solution is 



(y 2 /a)* Be-** (x - c) cos (kt - fax + fac - 7r/4), 
where A and B have the same values as before. It is noteworthy that A 

* The theory is due to Stokes, Papers, vol. m, p. 1. See Lamb's Hydrodynamics, p. 586. 
t Proc. Eoy. Soc. London, A, vol. xciv, p. 137 (1918). 
t G. Green, Phil Mag. (7), vol. in, p. 784 (1927). 



Fluctuating Temperatures 215 

and B are independent of a and that when the expressions in these solutions 
are integrated with respect to a from to oo the physically correct solution 
for the case of the instantaneous generation of a quantity of heat q at the 
origin at time t == is obtained in the form 



EXAMPLES 



1. In the problem of the oscillating plane the viscous drag exerted by the fluid is, per 
unit area, 

. / 1 /717\ 

(sin at cos at). 

[Rayleigh.] 

2. Discuss the equations 



( V -H -- ~5T \ 



a) =* a sin <^, (<^ a constant), 

where Q is a constant representing the angular velocity of the earth, and <f> is the latitude. 

[V. W. Ekman.] 

3-16. The solution (A of 3-15) may be generalised by regarding A 
as a function of /? and then integrating with respect to /S. 
A solution of a very general character is thus given by 

v = I V*te cos [3x - 



+ e -^ sin [jSa; - 2j/j3 2 *] e/r (j8) rfjS, 

Jo 

where (/> (j3) and ^ (j8) are arbitrary functions of a suitable character. 
Solutions of this type have been used by Rayleigh and by G. Green. 

Some useful identities may be obtained by comparing solutions of 
problems in the conduction of heat that are obtained by two different 
methods when the solution is known to be unique. 

For instance, if we use the method of simple solutions we can construct 
a solution 2jr 

t; = - S e t{ *- n -^ a /(f)df 

7Tn-oo Jo 

which is periodic in x with the period 2?!. 

When t = the series is simply the Fourier series of the function / (x) 
and the inference is that with a suitable type of function / (x) our solution 



216 Two-dimensional Problems 

is one which satisfies the initial condition v = / (x) when t 0. Now such 
a solution can also be expressed by means of Laplace's integral 



t; - 



rco 

e M f(t)d( 9 

J -co 

and this may be written in the form 



00 ft, --- 

2 e 4 *< 

JO 



71= oo 



When the order of integration and summation can be changed, a comparison 
of the two solutions indicates that , 

(x - _+ gftTT, 2 

2 e" 1 **-^-" 21 '* = (77/1/0* S 



This identity, which is due to Poisson, has recently, in tho hands of 
Ewald, become of great importance in the mathematical theory of electro- 
magnetic waves in crystals. The identity can be established rigorously in 
several ways : 

(1) With the aid of Fourier series. 

(2) By the calculus of residues. 

(3) By the theory of elliptic functions (theta functions). 

(4) By means of the functional relation for the f -function, Riemann's 
method of deriving this functional relation being performed backwards. 

An elementary proof based on the equations 

x 

n 



>e x astt->oo if #->#, ...... (1) 

2~2n n H j^^-ig-acz as n -^ oo if rn~^ -> x ...... (2) 

\n -h r) v ; 

has been given recently by Polya*. 

We have2 n ~ 1 < n\ for n= 1,2,3,... and so, forO<.r< 1, 

e** = 1 + ~ + ... < 1 + 2x + 2x* + ... = 15. 
1 ! 1 x 

Also, for 0< x< {, 

1 4- T Sr 3 7^ 3 

^J_ = 1 + 2x + 2x 2 + ~ < 1 + 2x 4- 2x 2 + ^ 



Therefore e -2x-x* < _~- < e -2* 

* G. P61ya, Berlin, Akad. Wiss. Ber. p. 158 (1927). 



Frisson's Identity 217 

On account of the symmetry of the binomial coefficients it is sufficient to 
prove (2) for r > 0. In this case . j. , 2\ / r 1\ 



. j. , 2\ / r 
n ( l ~ n) V ~ n) '" \ l ~ ~^ 



r/ 1W 2X / r-_l\ 

V n)\ n) \ n / 



V' 

the upper estimate in (B) having been applied. A use of the lower estimate 
gives an analogous result, which, on account of the fact that r 4 n~ 3 -> 0, com- 
pletes the proof of (2). 

Putting x = ZCD V = ze 2irtv ' 1 in the identity 



k / fyvn \ 

we obtain S [(V(za> v ) 4- 1 A/(z"")] 2m = 1 % ( , ) z l , 

-1<2<1 v--k \ m + W/ 

where k = [m/l] is the integral part of m/l. 

Now let s be an arbitrary fixed complex number and t a fixed real positive 
number. Putting I = V[( mt )]> z = gS ^ an( l dividing the series by 2 2m , we obtain 
the relation 



8P~\ 
2 coshM--~^n = 2 Jl + rAl^+ I 

[Vim] 



^ (C) 



Applying the limit (1) on the left and (2) on the right we finally obtain 

a + 2viv 

00 -jj-y CO 

V -00 00 

which is a form of Poisson's formula. 

To justify the limiting process which has just been performed in which the 
limit is taken for each term separately, it is sufficient to find a quantity inde- 
pendent of m which dominates each term in each series. 

There is little difficulty in finding a suitable dominating quantity for the 
terms on the right-hand side, but to find a suitable quantity for the terms on 
the left-hand side x>f the equation P61ya finds it necessary to prove the following 
lemma. Given two constants a and b for which a > 0, < b < IT, we can find 
two other constants A and B such that A > 0, B > and 

| cosh z | < e AxZ ~ Bv * 9 
when a < x < a, b<y<b and z = x + iy. We have, in fact, 

| cosh z | 2 = \ (1 + cosh 2x) - sin 2 */, 
but, for a < x < a, we have 

i <1 4- cosh 2x) < 1 -f- i 2 T^-T-r~ = 1 -f 2Ax 2 y say. 
n-i (2n)\ 



218 Two-dimensional Problems 

On the other hand, since sin y/y decreases as y increases from to TT, we have 
for - 6 < y < 6, 

^ > * = Vfff, 8 ay, V25>0. 

It follows from the inequalities that have just been established that 
| cosh z | 2 < 1 + 24z 2 - 2?/ 2 < e 2 <^ 2 -*" 2) , 

and this proves the lemma. 

To apply the lemma to the series (C) we note that in the first member 

| TTV/l | < 77/2, 

we therefore take b = 77/2. 

If s is real, the sum in the first member of (C) is dominated by the series 



I e -gr F-. 

V 00 

It is easy to dominate this series by one free from m. The case in which 

s is not real can also be treated in a similar manner. 

\ 

EXAMPLES 

1. If P = [ e~ x * [cos (xO) - sin (x9)] cos (2**0 2 ) d6, 

JO 

Q = / e -^ [cos (x^) + sin (xd)] sin (2 K ^ 2 ) d0, 

JO 
show by partial integration that 

xP=>-2 K tl Q , xQ = -1 K t*. 
ox ox 

Show also that, as t -> 0, 

4P-v4Q->rr*(^r i , 
and that consequently 

4P = 4C = 7r*(,cO~ > e- a;2 /4**. 

2. If C - ( e-^v cos (y 2 - a 2 ) dt/, 

7 - 

/ oo 

^f^ e-^ sin (y 2 - a 2 ) (^y, 
J -a 

prove that (7 + S = V(w/2), 0-^ = 2 I * e 29 * dd. [G. Green.] 

3-17. Conduction of heat in a moving medium. When the temperature 
depends on only one co-ordinate, the height above a fixed horizontal plane, 
and the vertical velocity of the medium is w, the equation of conduction is 

do 90_ d*e 

dt + V dy~ K ~dy*' ...... (A) 

where K is the diffusivity. When v is constant the equation possesses a 
simple solution of type 

/i + v\ = *A 2 , 



Conduction in a Moving Medium 219 

which may be generalised by summation or integration over a suitable set 
of values of /*. In particular, if we regard as made up of periodic terms 
and generalise by integration over all possible periods, we obtain a solution 

roo 

=, e ay [f( a ) sin (by -f- at) + g (a) cos (by + at)] da, 

.70" 

where 2*a = v w, bw = a, 



and / (a), g (a) are &c*itable arbitrary functions. The integral may be used" 
in the Stieltjes sense so that it can include the sum of a number of terms 
corresponding to discrete values of a. 

When v varies periodically in such a way that v = u (1 4- r COB at), 
where u, r and a are constants, a particular solution may be obtained by 



...... (B) 

where / (t) is a function which is easily determined with the aid of the 
differential equation. When v is an arbitrary function of t the equation 
(A) has a simple solution of type 



which may be generalized into 

= [ e w l -'""I ~ ** F(s)ds 

t J oo 

where F(s) is a suitable arbitrary function of s. 

The-solution (B) has been used by McEwen* for a comparison of the 
results computed from theory with the results of a series of temperature 
observations made off Coronado Island about 20 miles from San Diego in 
California. The coefficient K is to be interpreted as an "eddy conductivity" 
in the sense in which this term is used by G. I. Taylor. This is explained 
by McEwen as follows : 

At a depth exceeding 40 metres the direct heating of sea water by the 
absorption of solar radiation is less than 1 per cent, of that at the surface. 
Also, the temperature range at that depth would bear the same proportion 
to that at the surface if the variation in rate of gain of heat were due only 
to the variation in this rate of absorption. The direct absorption of solar 
radiation cannot then be the cause of the observed seasonal variation of 
temperature, which amounts to 5 C. at a depth of 40 metres and exceeds 
1 at a depth of 100 metres. Laboratory experiments show, moreover, 

* Ocean Temperatures, their relation to solar radiation and oceanic circulation (University of 
California Semicentennial Publications, 1919). 



220 Two-dimensional Problems 

that the ordinary process of heat conduction in still water is wholly in- 
adequate to produce a transfer of heat with sufficient rapidity to account 
for the whole phenomenon. It is now generally recognised that a much 
more rapid transfer of heat results from an alternating vertical circulation 
of the water in which, at any given instant, certain portions of the water 
are moving upward while others are moving downward. The resultant 
flow of a given column of water may be either upward or downward, or 
may be zero. The motion may be described as turbulent and a vivid picture 
of the process may be obtained by supposing that heat is conveyed from 
one layer to another by means of eddies. This complicated process produces 
a transfer of heat from level to level which, when analysed statistically, 
will be assumed to be governed by the same law as conduction except that 
the "eddy conductivity" or "JMischungsintensitat " will depend mainly 
on the intensity of the circulation or mixing process. 

An equation which is more general than (A) has been obtained by 
S. P. Owen* in a study of the distribution of temperature in a column of 
liquid flowing through a tube. 

Assuming, as an inference from Nettleton's experiments, that the shape 
of the isothermals is independent of the character of the flow, Owen con- 
siders an element of length 8?y fixed in space and estimates the amounts 
of heat entering and leaving the element across its two faces perpendicular 
to the y-axis to be 

*{-*%+''* 
and ^{-*^(0 + ^ 

respectively, where A is the area, p the perimeter of the cross-section of 
the tube, 9 the temperature of the element, E the emissivity, k, p and s 
the thermal conductivity, density, and specific heat of the liquid respec- 
tively, and where is the temperature of the enclosure which surrounds 
the tube. 

Owen thus obtains the equation 



A k * ~ psv y 8y - E P( - *o) % = Aps Sy, 

2 n-f-?('-*o)=4' 

dy* dy Aps { ' dt* 
where a 2 - k/ps. 

EXAMPLES 

1. Prove that a temperature which satisfies the equation 

de de d z e 

Ht ri~ = K 3^* ' 

and the conditions ^ 

= when z, = 0, 0, when y * 6, . when f - 0, 
* Proc. London Math. Soc. vol. xxm, p. 238 (1925). 



Wave Propagation by an Electric Cable 221 

is given by the formula 



ttn z * z *!b*) + (i?/4)}M. 
[Somers, Proc. P%$. &>c. London, vol. xxv, p. 74 (1912); Owen, foe. ci7.] 



2. If in the last example the receiver is maintained at a temperature which is a periodic 
function of the time, so that the condition 9 = & l when y b is replaced by d = 6 cos a>t 
when y = b, the solution is 

B = aev(i/-&)/2* (cosh 2nb cos 2w6) -1 [(cos m cosh wiy cos my cosh TI) cos o> 
(sin wi| sinh 7117 sin m^ sinh T?) sin a>t] 

4- 2e&~ 2 2 ( 

p-1 
where 



= {(v/2 K )* -f (a;/^) 2 } {i tan- 1 (4^a./ ? ; 2 )}. 
7i sin 

3-18. Theory of the unloaded cable. Consider a cable in the form of a 
loop (Fig. 13) having an alternator A at the sending end and a receiving 
instrument B at the receiving end. 
We shall suppose that the alter 
nator is generating a simple periodic 
electromotive force which may be 
represented as the real part of the 

expression Ee lni , where E and n are constants. Naturally, we arc interested 
only in the real part of any complex quantity which is used to represent 
a physical entity. 

Now, if CSx is the capacity of an element of length 8x with regard to 
the earth, the capacity of a length ox with regard to a similar element in 
the return cable must be ^Cox. Hence, if 7 is the current in the alternator 
and V Q the potential difference of the two sides of the cable at the sending 

end ' lc aF a/ 

~* C W~~~ ~dx' 

Now VQ is the difference between the generated electromotive force 
Ee int and the drop in voltage down the alternator circuit and a capacity 
CQ in series with it, consequently we have the equation 






+ F = 



Assuming that / can be expressed as the real part of X (x) e int and 
that / = / at the receiving end, we find on differentiating the last equation 
with respect to t and multiplying by C , 

(1 - C Q L n* + inC Q R Q ) 7 -f C - 



222 Two-dimensional Problems 

Hence the boundary condition at the sending end is 



- 

ox 

where h Q C = 0(1- C L n* -f inC R ). 

Similarly, if /j is the current at the receiving end and if the receiving 
apparatus is equivalent to an inductive resistance (L 19 RJ in series with a 
capacity C l , we have the boundary condition 

9/1 - 



where A^ = C (1 - C^n 2 + 

Assuming that there is no leakage, the differential equation for /is * 



and if X = K 1 cos /z (I x) + K 2 sin /z (I x), 

where I is the distance between the alternator and receiving instrument, 
and K l9 K 2 are constants to be determined, we have 

/x 2 = C (n*L - inR). 
Writing /x = a -f i/3, where a and /J are real, we have 

a 2 - )8 2 = LCn 2 , 2ap = - CRn, 
and so, if R 2 -f n*L 2 = G 2 , we have 

22 =Cn(G + nL), 2^8 2 - Cn (G - nL). 
When nL is large in comparison with R we may write 

and we have 

a = 

the wave- velocity being (CL)~^. In this case the wave-velocity and 
attenuation constant are approximately independent of the frequency, 
consequently a wave-form built up from waves of high frequency travels 
with very little distortion. 

The constants JK 1 and K 2 are easily determined from the boundary- 
conditions and we find that 



where F = (/^/^ 4/I 2 ) sin pi + 2/* (h + k^ cos pi. 

When E = the differential equation possesses a finite solution only 
when F = and this, then, is the condition for free oscillations. The roots 
of the equation F = 0, regarded as an equation for n, are generally complex. 

* Our presentation is based upon that of J. A. Fleming in his book, The propagation of electric 
currents in telephone and telegraph conductors. 



Roots of a Transcendental Equation 223 

This may be seen by considering the special case when (7 = C l = oo. This 
means that there are short circuits in place of the transmitting and re- 
ceiving apparatus. 

We now have A = A x = 0, p? sin [d = 0, and if we satisfy this equation 
by writing p,l SIT, where s is an integer, the equation 
sW = pip = l*C (n*L - inR) 

gives complex values for n. 

When J? = R l = R the roots of the equation for n are all real. This 
may be proved with the aid of the following theorem due to Koshliako v * . 

m n 

Let < -f iiff - 2 m s log (z - z 8 ) - S k 8 log (z - s ) 

5-1 S-l 

be the complex potential of the two-dimensional flow produced by a 
number of sources and sinks, the sources being all above the axis of x and 
the sinks all on or below the axis of x. 

Writing z s = a B + ib s , , = ,- iij, , 

where a 8 , b 8 , g s , r} 3 are all real, we shall suppose that 

b s > 0, T] S > 0, m s > 0, k s > 0. 
Now suppose that when x is real and complex 

n Ir' _ y ^w*t 

n , r-fe -/<*> + v(). 

S-l I*' fcj * 

where / (x) and (x) are real when x is real. If we superpose on the flow 
produced by the sources and sinks a rectilinear flow specified by the stream- 
function fa = x y tan o>, the stream -function of the total flow is 
^ = i/j Q -f fa and the points in which a stream -line iff = 6 cuts the axis of 
x are given by the transcendental equation 



or g (x) cos (x - d) + f (x) sin (x - 0) = 0. 

We wish to show in the first place that the roots of this equation are 
all real. Writing 



G (x] - iF (x 

IT (X) - lit (X = _ , . 

s l ^ bs *Vs/ 

we have G (a:) -f- iF (x) = e z (x - d) [/(*?) + *V ()], 

(a?) - *T (x) = e' <-*> [/ (x) - t^ (a?)], 
^ () = / () sin (a; ^ ) + gr (a:) cos (x - 0), 
G(x)=f (x) cos (a: - 6) - g (x) sin (a; - 6). 

* Mess, of Math. vol. LV, p. 132 (1926). Koshliakov considers only the case m, - 1, k, = 
without any hydrodynamical interpretation of the result. 



224 Two-dimensional Problems 

Hence, if z = x 4- iy is a root of the equation F (z) 0, we have for 
this root % 

Now let M^ and M 2 be the moduli of the -expressions on the two sides 
of the equation, then the equation tells us that M^ = J/ 2 2 , but 



M * _ e - 

1 ~ e 



and from these equations it appears that y > 0, M-f < M 2 2 , while if 
y < 0, M ! 2 > J/ 2 2 . Hence we must have y = 0, and so the roots of the 
equation ^ T (2) = are all real. 

Let us next determine the effect on the roots of varying the value of 6. 

If a; is a real root of the equation F (x) = 0, we have 

(dx/d9) [/' (x) sin (x - 9) + g' (x) cos (x - 0) -f- G (x)] - G (x), 

, . cos (x 0) sin (x 6) 1 

f(*\ r^i T^T'i' 

therefore (<te/rf0) [/ (a?) y' (x) - f (x) g (x) + {(7 (a;)} 2 ] - [6 y (a-)] 2 . 
Now r ; , -!,,-= S ( * ., - - 

i/'yi I Q rt I 'Y\ .\ / y ATT *? /i i* 

/ it*/ 1 "f~ t-j/ (.</ / ^ _a j \</ tt'g ^^5 ^ 

/ (a:) ig (x) s =i\ x ~ a s + ^5 ^ 
therefore 

-/(*) > 



The right-hand side is clearly positive and so dxfdO is positive for all 
real values of 6. This means that when x increases, the point in which the 
stream -line meets the axis of x moves to the right (i.e. the direction in 
which x increases). 

If we increase 6 by JTT, F (x) is transformed into G (x), and if we add 
another 77- to 0, the function G (x) is transformed into F (x), conse- 
quently we surmise that the roots of F (x) = are separated by those of 
G (x) = 0. To prove this we adopt Koshliakov's method of proof and 
calculate the derivative 

d F (x) J (x) g'J^l^f (x) g (x) + [F (x)]* + [G (*)] 
dxG(x)~ "" ~[~ 



This is clearly positive for all values of x and infinite, perhaps, at the 



Koshliakov *s Theorem 225 

roots of O (x) = 0. It is clear from a graph that the roots of F (x) are 
separated by those of (x) = 0, for the curve 



consists of a number of branches each of which has a positive shape. 

In Koshliakov's case when k s 0, m s = 1 the functions / (x) and g (x) 
are polynomials such that the roots of the equation / (x) -f ig (x) are 
of type z s = a s -f- ib s , where b s > 0. The associated equation F (x) = is 
now of a type which frequently occurs in applied mathematics. In par- 
ticular.if /(;;) = &&-*, p (*) = (ft + ft) a, 

the roots of the equation / (x) -f ig (x) = are ifa and i/3 2 and so we have 
the result that if j8 x and /? 2 are both positive, the roots of the equation 
(&L + &) x cos (x - 6) -f (&&> - # 2 ) sin (a: - 0) = ...... (B) 

are all real and increase with 6. 

The theorem may be applied to the cable equation by writing this in 
the form > , , 

7 _ 2 /^ fro Vl ~ M 2 (\ + A l)} 

^ (r ~ 



where y (7 = O, y^C^ = C, L == 

Now 



t- n - 

= (y + 2t> - 

and when the expression on the right is equated to zero, the resulting 
algebraic equation for p has roots of type a -t- ib, where b is positive, hence 
Koshliakov's theorem may be applied and the conclusion drawn that yil 
is real. Since in the present case /x 2 = CLn 2 , the corresponding value of 
n is also real. 

When = 0, x = wl, /?, = j8 2 = ZA, the equation (B) becomes identical 

with the equation 7 7 , -,. . 7 ,~ x 

2/o> cos o>& (a)' 2 /i^) in o>^, (C) 

which occurs in the theory of the conduction of heat in a finite rod, when 
there is radiation at the ends, into a medium at zero temperature. 
The equations of this problem are in fact 

dv S^v 
di ~~ dx 2 ' 
v=--f (x), for t = 0, 

~ + hv = at x = 0, =- + hv = at x = I, 
ox ox 

and are satisfied by 

v = e- K< 2t [A cos a>x -f B sin o>#] 

if - wB + hA - 0, 

a> (B cos a)l A sin o>Z) + h(B sin toZ -f -4 cos o>Z) = 0. 

B 15 



226 Two-dimensional Problems 

Eliminating A/B the equation (C) is obtained. The problem is finally 
solved by a summation over the roots of this equation, the root w = 
being excluded. 

Equations similar to (A) occur in other branches of physics and many 
useful analogies may be drawn. In the theory of the transverse vibrations 
of a string we may suppose that the motion of each element of the string 
is resisted by a force proportional to its velocity*. The partial differential 
equation then becomes 



- 
~ 



which is of the same form as (A) if K = RjL, c 2 = l/LC. 

An equation of the same type occurs also in Rayleigh's theory of the 
propagation of sound in a narrow tube, taking into consideration the 
influence of the viscosity of the mediumf. 

Let X denote the total transfer of fluid across the section of the tube 
at the point x. The force, due to hydrostatic pressure, acting on the slice 
between x and x + dx, is 

dp . 2 , d*x 

8 -if- ax = a 2 pdx -*<, , 
dx r dx 2 

where 8 is the area of the cross-section, p is the pressure in the fluid, p is 
the density and a is the velocity of propagation of sound waves in an 
unlimited medium of the same material. 

The force due to viscosity may be inferred from the investigation for 
a vibrating plane ( 3-15), provided that the thickness of the layer of air 
adhering to the walls of the tube be small in comparison with the diameter. 
Thus, if P be the perimeter of the inner section of the tube and F the 
velocity of the current at a distance from the walls of the tube, the tan- 
gential force on a slice of volume Sdx is, by the result of (3-15, Ex. 1), 
equal to 



where n/27r is the frequency of vibration. 

o v- 

Replacing VS by -~r we can say that the equation of motion of the 

ut 

fluid for disturbances of this particular frequency is 



- 

or S 



, 

= a 2 ---, . 
cte 2 

* Rayleigh, Theory of Sound, vol. I, p. 232. t Ibid. voL n, p. 318, 



Air Waves in Pipes 227 

This equation has been used as a basis for some interesting analogies 
between acoustic and electrical problems*. We shall write it in the abbre- 
viated form 



- 

dt* dt ~ dx*' 

Rayleigh's equation has been used recently by L. F. G. Simmons and 
F. C. Johansen in a discussion of their experiments on the transmission of 
air waves through pipesf. 

At the end x = the boundary condition is taken to be 

X = X sm(nt), ...... (D) 

and a solution is built up from elementary solutions of type 

X = Ae tntmx , 
where a 2 ra 2 = Hn 2 -f iKn. 

Since m is complex, we write m = a 4- ip. A solution appropriate for 

o y- 

a pipe of length I with a free end (x = I) at which x - = is 

X = A {e~ ax sin (nt - px) + e-**' sin (nt - fix')} 

+ C {e-** cos (nt - px) -f e~ ax ' cos (nt - px')}, 

where x f = 21 x, and where the constants A, C are chosen so that 
X = A {1 + e- 2 *' cos 2 pi} -f Ce- sin 2pl y 

= - Ae-** 1 sin 2pl + G {1 4- e~* al cos 2 pi} . 
These equations give 

^r = (1 + e~ cos 2pl) X Q , G - e-* 1 sin 2pl . Z , 
where T = 1 -f 2e~ 2aZ cos 2j8/ -f e~^ 1 . 

In the case of a pipe with a fixed end the boundary condition is X = 
at # = Z, anc we write 

X = A [>-* sin (w< - px) - e-**' sin (^ - px')] 

-f (7 [e~ aa! cos (nt - #) - e*' cos (n^ - j8a?')]- 
The boundary condition (D) is now satisfied if 

Z = A {1 - e- 2 *' cos 2pl} - Ce~ cos 2j8, 
= Ae~ sin 2j8Z + G {1 - e- 2 *' cos 



Therefore 

G^^L = (1 - e-** cos 2^) Jf 09 GC7 = - X Q e-** sin 2)81, 
where G = 1 - 2e" 2ttZ cos 2^8Z -f- e^ 1 . 

* See a recent discussion by W. P. Mason in the Bell System Technical Journal, vol. vi, p. 258 
(1927). 

t Advisory Committee far Aeronautics, vol. n, p. 661 (1924-5) (E.-M. 957, Ae. 176). 

15-2 



228 Two-dimensional Problems 

If y denotes the ratio of the specific heats for air, the pressure at any 
point exceeds the normal pressure p by the quantity 



where X = S. The excess pressure at the fixed end is consequently 

P-Po = 2&~" YPo ( 2 + )* ~ l sin ( nt ~ & + #). 
. . , pA + aC 

where tan ^ = a^ T^' 

The following conclusion is derived from a comparison of theory with 
experiment : 

"Marked divergence between observed and calculated results shows 
that existing formulae relating to the transmission of sound waves through 
pipes cannot be successfully employed for correcting air pulsations of low 
frequency and finite amplitude." 

3-21. Vibration of a light string loaded at equal intervals. In recent 
years much work has been done on methods of approximation to solutions 
of partial differential equations by means of a method in which the partial 
differential equation is replaced initially by a partial difference equation 
or an equation in which both differences and differential coefficients appear. 
Such a method is really very old and its first use may be in the well-known 
problem of the light string loaded at equal intervals. This problem was 
discussed by Bernoulli* and later in greater detail by Lagrange|. 

Let the string be initially along the axis of x and let the loading masses, 
which we assume to be all equal, be concentrated at the points 

x = na, n = 0, 1, 2, ____ 

Let y n be the transverse displacement in a direction parallel to the 
/-axis of the mass originally at the point na, then if the tension P is re- 
garded as constant, we have for the motion of the nth particle 

amy n = P (y n+l - y n ) + P (y n _^ - y n ). 
Writing k 2 am = P, the equation becomes 

tin - & (y n+1 + y n _i - 2y n ). ...... (A) 

Let us now put u zn = y n , u 2n+1 = k(y n - y n+1 ), 
then ii 2n = k (u 2n ^ - u 2n+1 ), 

or, if s is any integer, 

* Johann Bernoulli, Petrop. Comm. t. m, p. 13 (1728); Collected Works, vol. in, p. 198. 
t J. L. Lagrange, Me'canique Analytique, 1. 1, p. 390. 



Vibration of a Loaded String 229 

This is a difference equation satisfied by the Bessel functions and a 
particular solution which will be found useful is given by* 

u s = AJ 8 ^ (2kt), 
where A and a are arbitrary constants and 

oo (__\s (l~\m+2s 

'M-.?.*-.-! nff.+ D <B) 

Let us first consider the ideal case of an endless string and suppose that 
initially all the masses except one are in their proper positions on the axis 
of x and have no velocity, while the particle which should be at x = na 
has a displacement y n = ^ and a velocity y n = v, then the initial conditions 

are 7 7 

u 2n = v, u 2n+l - 77, u 2n _i = - ??, 

while u s is initially zero if s does not have one of the three values 2n 1, 
2/r, 2n -f 1. A solution which satisfies these conditions is 

u s = v J S _ 2M (2fc) + fey [J s - 2n -i (2fo) - e/ 5 _ 2n+1 (2fa)L 

for, when t = 0, J T (2fe) is zero except when r = and then the value is 
unity. 

When all the masses have initial velocities and displacements the 
solution obtained by superposition is 

u s = Xv n J s _ 2n (2kt) -f k^ n [J^^ (2kt) - J 8 _ 2n+l (2kt)] (C) 

If v n = we find by integration that 

y s = Si, n J 2s _ 2n (2fcJ). (D) 

Let us now discuss the case when this series reduces to one term, 
namely, the one corresponding to n = 0. Referring to the known graph 
of the function J 2s (2kt), to known theorems relating to the real zeros and 
to the asymptotic representation f 

J 28 (2kt) = (7r&)-*cos(2 to - -^-j- ). (E) 

we obtain the following picture of the motion : 

The disturbed mass swings back into its stationary position, passes 
this and returns after reaching an extreme position for which | y | < ^ . 
Its motion always approaches more and more to an ordinary simple 
harmonic motion with frequency initially greater than &/TT, but which is 
very close to this value after a few oscillations. The amplitude gradually 
decreases, the law of decrease being eventually (rrkt)"^. This diminution 

* T. H. Havelock, Phil. Mag. (6), vol. xix, p. 191 (1910); E. Schrodingor, Ann. d. Phys. 
Ed. XLIV, S. 916 (19H); M. Koppo, Pr. (No. 96) Andreas- Realgymn. Berlin (1899), reviewed 
in Fortschritte der Math. (1899). 

f Whittaker and Watson, Modern Analysis, p. 368. The formula is due to Poisson. An ex- 
tension of the formula is obtained and used by Koppe in his investigation. The complete asymptotic 
expansion is given in Modern A nalysis. 



230 , Two-dimensional Problems 

depends on the fact that the vibrational energy of the mass is gradually 
transferred to its neighbours, which part with it gradually themselves and 
so on along the string in both directions. After a long time, when 2kt is 
so large that the asymptotic representation (E) can be used for the 
Bessel functions of low order, the masses in the neighbourhood of the 
origin vibrate approximately in the manner specified by the "limiting 
vibration" of our arrangement, neighbouring points being in opposite 
phase. The amplitudes, however, decrease according to the law mentioned 
above and the range over which this approximate description of the 
vibration is valid gets larger and larger. 

According to the formula (D) all the masses are set in motion at the 
outset, and all, except the one originally displaced, begin to move in the 
positive direction if T? O > 0. 

Let us consider the way in which the mass originally at x = na begins 
its motion. The larger n is, the slower is the beginning of the motion and 
the longer does it continue in one direction. This is because J 2n (2kt) 
vanishes like A n t 2H , as / approaches zero, A n being the constant multiplier 
in the expansion (B). Also because the first value oi 2kt for which the func- 
tion vanishes lies between ^/{(2ri) (2n -f 2)} and V{(2) (% n + *) ( 2 ^ + 3 )}. 

It is interesting to note that in this elementary disturbance there is 
no question of a propagation with a definite velocity c as we might expect 
from the analogous case of the stretched string. Let us, however, examine 
the case in which all the particles are set in motion initially and in such a 
way that the resulting motion is periodic. 

Writing y n = Y n e 2lku>t we have the difference equation 



If a) = sin <f> this equation is satisfied by 

Y n = A sin 2n<f> + B cos 2n<f>. ...... (F) 

Choosing the particular solution 

Y = Ce~ 2in * 

* n ^ c 9 

we have y n = Ce 2i(ktB{n *~ n * } . 

Making kt sin < n<j> constant we see that the phase velocity is 

_ ak sin <f> 

-? 

The period T is given by the equation 

T = __ 
k sin </> ' 

and the wave-length A by the equation 

A = cT ira/<f>. 



Group Velocity 231 

The phase velocity thus depends on the wave-length and so there is a 
phenomenon analogous to dispersion. Introducing the idea of a group 

velocity U such that ~ x ^, 

3A r/ 9A 

3* + U dx~^ 

that is, such that A does not vary in the neighbourhood of a geometrical 
point travelling with velocity U, we next consider a geometrical point 
which travels with the waves. For this point A varies in a manner given 

by the equation * ^ ~^ ~ , a 

^ ^A _ x dc _ , dc cc 

a7 + C fo fo 5Aai' 

the second member expressing the rate at which two consecutive wave- 
crests are separating from one another. Eliminating the derivatives of A 
we obtain the formula of Stokes and Rayleigh, 

Z7 = c-A~=tf(A),say: 

In the present case 

U = ak cos (7ra/A) = ak cos <f>. 

When A-> oo, U -> ak = U(co) = c(oo). 

Hence for long waves the group velocity is approximately the same as 
the wave velocity. For the shortest waves <f> = \TT, we have U = 0. 

When there are only n masses the two extreme ones being at distance 
a from a fixed end of the string, the equations of motion are 

& + & (2ft - - ft) = 0, 
$2 + & (2y 2 - ft - ft) = 0, 



Assuming y s = Y 8 e 2ik<at as before and eliminating the quantities Y 8 
from the resulting equations we obtain the following condition for free 
oscillations : 

2 cos 2<f> - 1 | = 0, 

- 1 2 cos 2^ 1 
- 1 2 cos 2<j> - 1 

t 

where there are n rows and columns in the determinant. Since 



it is readily shown by induction that 



n ~ sin 2<f> 
* See Lamb's Hydrodynamics, p. 359. 



232 Two-dimensional Problems 

This is zero if 2 (n -f- 1) (/> = rvr, r = 1, 2, ... n; we thus obtain /i different 
natural frequencies of vibration. When the motion corresponding to any 
one of these natural frequencies is desired we use an expression of type 
(F) for Y n and the end condition Y n = will be satisfied by writing B = 0. 
Hence one of the natural vibrations is given by 

y m = A sin2w<e 2i *< 8ln *, 

where 2 (n + 1) (f> = TTT (r = 1, 2, ... n). 

If the velocity is initially zero we write 

y s = ^4 sin 2s<^ . cos (2& sin <f>). 

Let us examine more fully the case in which n = 2. The possible values 
of (f> are and -- , consequently in the first case 



A sin ^ cos (2Kt sin ^ j , y 2 = A sin ~ cos ( 2kt sin ~ j ; 



y l and j/ 2 have the same sign and the string does not cross the axis of x. 
In the second case 

A . 2?! / rt/J . 77\ ,.477 / rt/J . 77 

y l = A sm cos ( 2tct sin - , s/ 2 = A sin cos ( 2kt sm - 

o \ o/ o \ o 

^ and y 2 have opposite signs, the string crosses the axis of z at its middle 
point which is a node of the vibration. 

When n = 3 we find in a similar way that there is one vibration without 
a node, one with a node and one with two nodes. 

The extension to the case in which n has any integral value is clear. 
The general vibration, moreover, is built up by superposition from the 
elementary vibrations which have respectively 0, 1, 2, ... n 1 nodes, the 
nodes of one elementary vibration such as the 5th being separated by those 
of the (s - l)th. 

If we regard this solution as valid for all integral values of s, we may 
apply it to the infinite string. The initial value of y a is now A sin 2s<f> and 
so by applying the general formula we are led to the surmise that there is 
a relation 

00 

sin 2s<f> . cos (2kt sin cf>) == S sin 2p<f> . J 2 s-2 (2&0> 

J>= 00 

which is true for all real values of <f>. This relation is easily proved with 
the aid of well-known formulae. 

An equation similar to (A) occurs in the theory of the vibrations of a 
row of similar simple pendulums (a, m) whose bobs are in a horizontal 
line and equally spaced, consecutive bobs being connected by springs as 
shown in Fig. 14. Using y n to denote the horizontal deflection of the nth 



Electrical Filter 233 

bob along the line of bobs and supposing that the constants of the springs 
are all equal, the equations of motion are of type 

/ / x 7 / x my //-.x 

tYlij == K (I/ I/ } rC ( I/ 11 ) I/ (\JTI 

The periodic solutions of this equa- 

tion give a good illustration of the filter 
properties of chains of electric circuits 
that were discovered by G. A. Camp- 
bell*. The mechanical system may, in 
fact, be regarded as an analogue of the Flg> 14 ' 

following electrical system consisting of a chain of electrical circuits each 
of which contains elements with . 

inductance and capacitance (Fig. I ' . . 

15). _- v/n-i r (*n T V^TIM 

The following discussion is ' ' ' 

based largely upon that of T. B. Fig ' 15 ' 

Brown f. When the chain is of infinite length and the motion is periodic 

$n appropriate solution is obtained by writing 

y n = Ar~ n sin (pt n<f>). 

If mg ap 2 = 2akQ the equations for r and (f> are 

(1 - r 2 ) sin <f> = 0, (r 2 + 1) cos <f> = 2r (1 + Q). 

These are satisfied by <f> = 0, r ^ 1 and by r = 1, cos <f> = 1 + Q. In the 
latter case there is transmission without attenuation but with a change of 
phase from section to section, the phase velocity corresponding to a fre- 
quency / = p/2n being v ^px = Zirfx 

(h (b 

where x is the length of each section. This type of transmission is possible 
only when Q lies between and - 2, that is when/ lies between / t and/ 2 , 
where , . I/AJ \ 

/ / fl\ I 4/c Q \ 

O f I i & 1 9^-f / I _L y I 

ZTT/I = A / I - I , ^^[/2 ~~ A / I ' " I 
J1 V \aJ V \w a/ 

This range of frequencies gives a pass band or transmission band. On 
the other hand, when < = we have r = 1 -f Q [(1 + Q) 2 - 1]*, and it 
is clear that r > when /< f l9 r < when /> / 2 . The negative value of 
r indicates that adjacent sections are moving in opposite directions with 
amplitudes decreasing from section to section as we proceed in one direction 
down the line. We may use a positive value of r if we take <f> = TT instead 
of <f> = 0. 

It should be noticed that r is real only when / lies outside the pass 

* U.S. Patent No. 1,227,113 (1917); Bell System Tech. Journ. p. 1 (Nov. 1922). 
t Spurn. Opt. Soc. America, vol. viu, p. 343 (1924). 



234 Two-dimensional Problems 

band. There are two regions in which / may lie and these are called stop 
bands or suppression bands ; one of these is direct and the other reverse. 
The stopping efficiency of each section is represented by log e | r \ and this 
is plotted against /in Brown's diagram. 

For a further discussion of wave-filters reference may be made to 
papers by Zobel, Wheeler and Murnaghan. 

Let us now write equation (G) in the form 

(D 2 + c 2 ) y n - & 2 (y M + y n-1 ) f 

where b*= k , c 2 = ~+?, D=~ 4 , 

m ma at 

and let us seek a solution which satisfies the initial conditions 

2/0=1, * 
2/o = 0* ?/i 

One way of finding the desired solution is to expand y n in ascending 
powers of 6 2 . Writing 

y n = (n, n) b* n + (n,n + 2) 
it is found by substitution in the equation that 

26 2 \ n (/&_+_!) (n + 2) /_J 26 2 
~ ~ 



1 ["/ 26 2 \ n 
2/71 ~~ 2 [U> 2 + c 2 / 



_ 

2 . 4(2n + 2)T2rT+ 4) 

the law of the coefficients being easily verified. The meaning which must 
be given to 2 



is one in which the Taylor expansion of the expression in powers of t starts 
with t 2m . (2b 2 ) m /(2m) I 

An expression which seems to be suitable for our purpose is obtained 
by writing 

cos ct = 75 ezt ~*- 



where C is a circle with its centre at the origin and with a radius greater 
than c. The result of the operation is then 

2b 2 m I zdz 



and we obtain the formal expansion 

- I _1_ f *t _ l dz _ [7 262 V 4. (+ 1 )( ra + 2 ) /_ 2 ^ a _V^ , I 

y " 2" ' 2 w i J c e z 2 + c 2 LV3 2 + cV 2 (2 + 2) U a + cV ;' + '"J ' 



Torsional Vibrations of a Shaft 235 

In particular, 

1 r e zt zdz 

~ 



e zt zdz 



1 r 

2/1 = 2*ri ] e [> 2 c~ 464}* z 2 + c 2 -f [(z 2 +c 2 ) 2 - 
and generally 



1 r 
" ^ 2m j 



2fe 2 



! c [(z 2 + c 2 ) 2 - 46 4 ]i (z 2 + c 2 + [(z 2 + c 2 ) 2 - 46*]*) * 

It is easy to verify that this expression satisfies equation (G) and the 
prescribed initial conditions. 

When c 2 = 26 2 the formula reduces to 

!_ f e" _ \ b ___ 



which must be equivalent to J 2n (2bt). 
The solution of the equation 

(D + c 2 ) y w = 6 2 (^^ + y,^), 
which satisfies the initial conditions 

2/o = 1, 2/i = 0, t/ 2 = 0, ... when t = 0, 
is given by the formula 

= ^ f _e?*zdz 



An equation which is slightly more general than (A) occurs in the theory 
of the torsional vibrations of a shaft with several rotating masses*. This 
theory can be regarded as an extension of that of 1-54 and as a preliminary 
study leading up to the more general case of a shaft whose sectional pro- 
perties vary longitudinally in an arbitrary manner. 

Let /!, / 2 , / 3 , ... /jv be the moments of inertia of the rotating masses 
about the axis of the shaft, 19 2 , 3 , . . . N the angles of rotation of these 
masses during vibration, k l9 k 2 , k 3 , ... k N the spring constants of the shaft 
for the successive intervals between the rotating masses. Then 

*i (*i ~ 2 ), ^2 (<?2 - 9*), -. *W (Vi - ON) 

are torque moments for these intervals. Neglecting the moments of inertia 
of these intervening portions of the shaft in comparison with / x , 7 2 , ... /# 
the kinetic energy T and the potential energy V of the vibrating system are 
given by the equations 

2T=/ 1 d 1 *+/A"+.../A > , 

2F = k, (0, - 2 )* + kt (0 2 - 3 ) 2 + - **-i (0.v-i - *)', 

* See S. Timoshenko, Vibration Problems in Engineering, p. 138; J. Morris, The Strength of 
Shafts in Vibration, ch. x (Crosby Lockwood, London, 1929). 



236 Two-dimensional Problems 

and Lagrange's equations give 

/A + k n (0 n - n+1 ) - k n _, (0^ - n ) = 0, 

Except for the two end equations, for which n = I, N, respectively, 

these equations are the same as those of a light string loaded at unequal 

'Intervals. When k^ ----- Jc 2 = & 3 = ... = k N the foregoing analysis can be used 

with slight modifications. A second case of some mathematical interest 

arises when A^ = Jfc 3 = fe 6 = ... fc 

7. __ 7* __ 7^ __ 7, 

^ 2 4 ~~ 6 ~~ '^* 

EXAMPLES. 

1. By considering special solutions of the equation of the loaded string prove that the 
following relations are indicated: 

00 

(n - x? = S 

7/1= -CO 



2. Prove that the equation 

J M 

^ = J [^_, (x) + ^ n+1 (x) - 2^ n (a:)] 
is satisfied by F n (x) = e^t*^) 2 7 n _ m (* - a) F m (a), 

m^-~<x> 

where / n (a-) = *~V n (ix), 

and obtain the solution in the form of a contour integral. 

3. Prove that the equation 

lln = V [y n+z -H 2/ n _ 2 
is satisfied by 

1 ( _ e 
?w/n " 2^J Jc [( Z a~+ c 2 ) 2 - 46*1 z 2 + c-f [(z 2 + c 2 ) 2 - 

4. Each mass in a system is connected with its immediate neighbours on the two sides 
by elastic rods capable of bending but without inertia. Assuming that the potential energy 
of bending is 

F = . 



prove that the oscillations of the system are given by an equation of type 



when r > 1 and obtain the two end equations. [Lord Rayleigh, Phil. Mag. (5), vol. XLTV, p. 356 
(1897); Scientific Papers, vol. rv, p. 342.] 

6. Prove that a solution of the last equation is given by 

Vr = J r Q C08 (Y "" !) * [Havelock.] 

3-31. Potential function with assigned values on a circle. Let the 
origin and scale of measurement be chosen so that the circle is the unit 



Potential with Assigned Boundary Values 237 

circle | z \ = 1 and let z' = e ie/ be the complex number for a point P' on 
this circle. Our problem is to find a potential function V which satisfies 
the condition T7 



as z ~* z - 

To make the problem more precise the way in which the point z 
approaches z' ought to be specified and something must be said about the 
restrictions, if any, which must be laid on the function/ (0'). These points 
will be considered later; for the present it will be supposed simply that 
/ (0') is real and uniquely defined for each value of 0' when 9' is a real angle 
between TT and rr. The mode of approach which will be considered now 
is one in which z moves towards z' along a radius of the unit circle. In 
other words, if z = re!*, where r and 9 are real, we shall suppose that 9 
remains equal to 9' and that r -> 1. 

Now let z = r . e~ id be the complex quantity conjugate to z; an attempt 
will be made first of all to represent V by means of a finite or infinite series 

F = c + 2 (c n z n + c^i"), ...... (A) 

71-1 

where c is a real constant and c n9 c_ n are conjugate complex constants. 
When the series contains only a finite number of terms it evidently 
represents a potential function and in the limiting process z-+z' it tends 
to the value Ec n z' n , where the summation extends over all integral values 
of n for which c n ^ 0. Negative indices are included because z n -> z'~ n . 

Supposing now that the finite series represents the function/ (#'), the 
coefficient c n is evidently given by the formula 



for the integral of e im6 ' between -n and TT is zero unless m = 0, conse- 
quently the term c n z' n in the series for/ (#') is the only one which contributes 
to the value of the integral. 

A function/ (0') which can be represented as the sum of a finite number 
of terms of type c n e~ ine ' is evidently of a special nature and the natural 
thing to do is to endeavour to extend the solution which has just been found 
by considering the case in which an infinite number of the constants c n 
defined by the formula (B) are different from zero. The series (A) formed 
from these constants then contains an infinite number of terms. 

Let us now assume that the f unction /(#') is integrable in the interval 
TT < 6' < IT. Since the series 



1 + S r n [e*<*-*'> + e - in <'- 6 '>] == K (9 - 9') 
i 

is uniformly convergent for all points of this interval if | r | < 1, it may be 



238 Two-dimensional Problems 

integrated term by term after it has been multiplied by/ (0'). The potential 
function V may, consequently, be expressed in the form 

d0' 9 (C) 



where K (co) =1 + 2 r n cos na> = - ~ --- A . 

n-i 1 - 2r coso> -f r 2 

The integral representing our potential function F is generally called 
Poisson's integral and will be denoted here by the symbol P (r, 6) to 
indicate that it depends on both r and 6. This integral is of great importance 
in the theory of Fourier series as well as in the theory of potential functions. 

The formula (C) may be obtained in another way by using the Green's 
function for the circle. If P (r, 6) is the pole of the Green's function, 
Q (r- 1 , 6) the inverse point and P' (/, 6') an arbitrary point which is inside 
the circle (or on the circle) when P is inside the circle and outside (or on) 
the circle when P is outside the circle, an appropriate expression for the 
Green's function is 



where A is an arbitrary point on the circle. This expression is evidently 
zero when P' is on the circle, it becomes infinite in the desired manner when 
P' approaches P and it is evidently a potential function which is regular 
except at P. 
The formula 

" 2~7r J o F- 2r cos~(0 ~(F)+~r* 

represents a potential function which takes the value / (0) on the circle 
and is regular both inside and outside the circle. 
When r = the formula gives the relation 



where F is the value of F at the centre of the circle. This is the two- 
dimensional form of Gauss's mean value theorem. 

When / (6) is real for real values of 6 the formula (0) may be written 
in the form 



of the sum of two conjugate complex quantities each of which takes the 
value J/ (6) at the point z = e ie on the unit circle, and we deduce Schwarz's 
more general expression 



...... (D) 



Potential with Assigned Boundary Values 239 

for a function F (z) whose real part on the unit circle is/ (6). The imaginary 
part of F (z) is a potential function i W which is given by the formula 

w - h 4- 1 (**rsm(e-6')f(e') M' 
TrJo l-2reos(0-0') + r 2 ' 

If in the formula (D) we have / (2-n 6) = / (0) we obtain Boggio's 
formula for a function F (z) whose real part takes an assigned value / (0) 
on the semicircle z = e ie , < 9 < TT, and whose imaginary part vanishes 
on the line z = cos a, < a < TT, i.e. the diameter of the semicircle, 



1- 2zcos0'-f z 2 * 

When F = / (z), where / (z) is a function which is analytic in the unit 
circle | z' z \ 1, Gauss's mean value theorem may be written in the 
form 



and is then a particular case of Cauchy's integral theorem. By means of 
the substitution z' = pz, F(pz)--=f(z) the theorem may be extended to a 
circle of radius p. 

If on the circle we have | / (z f ), \ < M, the formula shows that | / (z) | < M. 

More generally we can say that if / (z) is a function which is regular 
and analytic in a closed region G and is free from zeros in G, then the 
greatest value of | / (z) \ is attained at some point of the boundary of G 
and the least value of | / (z) \ is also attained at some point on the boundary 
of G. In this statement values of / (z) for points outside G are not taken 
into consideration at all. 

If/ (z) is constant the theorem is trivial. If/ (z) is not constant and has 
its greatest value M at some point z inside G we can find a small neigh- 
bourhood of z entirely within G for which | / (z) \ < M , and if O is a small 
circle in this neighbourhood and with z as centre this inequality holds for 
each point of C and so 



which leads to a contradiction. The theorem relating to the minimum value 
of | /(z) | may be derived from the foregoing by considering the analytic 
function l//(z). 

3-32. Elementary treatment of Poissorfs integral*. To find the limit 

lim P (r y 0) 

r->l 

it will be assumed in the first place that / (9) is integrable according to 

* This treatment is based upon that given in Carslaw's Fourier Series and Integrals. 



240 Two-dimensional Problems 

Ricmann's definition and that if it is not bounded it is of such a nature 
that the integral 

f(9')dff 

j it 

is absolutely convergent. 

Let us now suppose that is a point of the interval TT < 6 < TT which 
does not coincide with one of the end points. We shall suppose further 
that the limit 



,. r , n , x /n 
hm [f(6+ r) + /(0- 



T->0 



/T ^ 
...... (E) 



exists and is equal to 2F (6), where F (9) is simply a symbol for a quantity 
which is defined by this limit when 6 is chosen in advance. No knowledge 
of the properties of the function F (9) will be required. 

Now let a function <D (#') be defined for all values of 6' in the interval 
( TT < 9' < TT) by the equation 

<i> (0') = f (0') - F (0). 
Then 



(0 - 9')[f(9') - F (9)] d9' 



P (r, 9) - F (9) = 



Since, by hypothesis, the limit (E) exists, a positive number 77 can be 
found so as to satisfy the conditions 

\f(0 + a)+f(0-a)-2F(0)\< , 
when < a < 17, 9 77 > 77, 9 + 77 < 77, 

e being an arbitrary small positive quantity chosen in advance. Then 



fo + y fi 

K (0 - 0') O (6') dO' = K (a) [O (0 + a) + (0 - a)] 
fl-7) JO 



da 



and so 



(9)] da, 



K(0-0')Q> (O 1 ) 



ri TT 

< \ K (a) da < e\ K (a) da = 

JO J-TT 

Also, when < r < 1, 

[' ~*/c (0 - 9') O (fl') dfl' + f * ic (ff - 0') (0') dfl 7 

-rr h + n 



277 



(77), say, 
where A is a positive quantity. 
But, when 0< r < 1, 



(1 



-f 4r sin 2 77/2 2r sin 2 ij/2 ' 



Discussion of Poissorfs Integral 241 

Hence 2irAK (T?) < 2-Tre if r is so chosen that 

Lz r__ 5 

2rsin 2 "7p < 2' 
and this inequality is satisfied if 

r> s 



Combining the two results we find that 

\P(r,B)-F (9) | < 2e, 

if !>,>[, + * sin.*]' 1 . 

Hence when the limit (E) exists, 

P(r, 0) -*jF(0) asr -> 1. 

When is a point of continuity of the function/ (0) we have, of course, 
F (0) = / (0) and so V tends to the assigned value. To prove that V is a 
potential function when r < 1 it is sufficient to remark that the series 



obtained by differentiating (A) term by term with respect to z is uniformly 

Y 

convergent for r<s< 1, where s is independent of r and 0, hence -g-- 

12 T/ 

exists and is a function of z only. The equation x ~- = then follows 

immediately. 

The behaviour of Poisson's integral in the neighbourhood of a point 
on the circle at which / (0) is discontinuous is quite interesting. Let us 
suppose that / (0) has different values f (0) and / 2 (0) when the point is 
approached along the circle from different sides, then if the point 9 is 
approached along a chord in a direction making an angle air with the 
direction of the curve for which increases the definite integral tends to 

the value , - /m f /m 

(1 -)A(0) + / 2 (0). 

A proof of this theorem is given by W. Gross, Zeits. f. Math. Bd. n, 
S. 273 (1918). When/ (0) is continuous round the circle we have the result 
that V -> / (0) as any point on the circle is approached along an arbitrary 
chord through the point. This theorem has also been proved by P. Pain- 
leve, Comptes Rendus, t. cxn, p. 653 (1891) and by L. Lichtenstein, Journ. 
f. Math. Bd. CXL, S. 100 (1911). 

EXAMPLES 
1 Show by means of Poisson's formula that if 



- 1 (0 < 6 < TT), 

16 



242 Two-dimensional Problems 

the potential V is given by the equation 

F = 1 + 2 tan- 1 ~ . . (r 2 < a 2 ). 
TT 2ar sin ' 

2. Let the unit circle z = e* d be divided into n arcs by points of division B t , 2 , . . . n , where 
< X < 6 2 < ... < n = 2*. Let <f> -f i^ = / (z) be analytic for | z | < 1 and let < satisfy the 
following conditions on the circle 

* = <W m _ l <6<0 mt c^-c^ 
c m being an arbitrary constant, then 

27T/ (zj - - 27r Cl 4- 2 (c m - c,) [0, + 2* log (e 1 ^ - z)]. 



[H. Villat, W/. rfe la Soc. Math, de 'France, t. xxxix, p. 443 (1911); "Aper9us the"oriques 
sur la resistance des fluides," Scientia (1920).] 

3-33. Fourier series which are conjugate. When r is put equal to 1 in 
the series (A) the resulting series may be written in the form 



n oo 

and is the " Fourier series" associated with the function/ (8). 

Separating the real and imaginary parts, the series may be written in 
the form ^ 

a -f S (a v cos v9 + b v sin vd), ...... (A') 



where * = J (0>) d0 ' > 

a v = - [ n f(6')coav0'd6', 

TT J -IT 

6 F = L [ 

T J 



The constants a,, 6,, are called the " Fourier constants" associated with 
the function/ (#'). In terms of these constants the series for V is 

00 

x 7 = a -f S r v (a,, cos v0 + 6^ sin i/0). ...... (B') 

v-l 

When all the coefficients are real the series for the conjugate potential 

W is 

6 -f- r (a sin - ^ cos 0) + r 2 (a 2 sin - 6 2 cos 0) + ... , ...... (C') 

and this is associated with the series 

6 -f (a x sin &! cos 0) + (a 2 sin b 2 cos 0) -f ..., ...... (D') 

which, when 6 = 0, is called the conjugate* of the Fourier series (A'). 
There is now a considerable amount of knowledge relating to the con- 
jugate series. One question of importance in potential theory is that of the 

* Sometimes it is this series with the sign changed which is called the conjugate series. See 
L. Fejer, Crelle, vol. CXLH, p. 165 (1913); G. H. Hardy and J. E. Littlewood, Proc. London Math. 
Soc. (2), vol. xxiv, p. 211 (1926). 



Conjugate Fourier Series 243 

existence of a function g (9) of which the foregoing series is the Fourier series. 
In this connection we may mention a theorem, due to Fatou *, which states 
that if/ (6) is everywhere continuous and the potential W is expressed in the 
form W W (r, 0), the necessary and sufficient condition for the existence 
of the limit 

lim W(r,0) = g(0) ...... (E') 

r-+l 

for any assigned value of 6 is that the limit 

lim f * [/ (0 + T ) -/ (8 -r)] cot ldr = - 2^(0) ...... (F) 

->OJe ^ 

should exist. Fatou has also shown that if / (6) has a finite lower bound 
and is such that / (0) is integrable in the sense of Lebesgue then the limit 
(F') exists almost everywhere. 

Lichtensteinf has recently added to this theorem by showing that the 
integral 

| 

J 7 

exists when ft/ 

J IT 

exists. 

For further properties of Poisson's integral and conjugate Fourier 
series reference may be made to the book of G. C. Evans on the logarithmic 
potential J and to Fichtenholz's paper in Fundamenta Mathematicae (1929). 

Fatou's expression for g (9), when/ (6) is given, is 

9 (6) = 277 



In the last integral the symbol P denotes that the integral has its principal 
value. Villat has deduced this expression by a limiting process with the aid 
of the result of Ex. 2, 3-32. The formula is quite useful in the hydro- 
dynamical theory of thin aerofoils. 

An alternative expression, obtained by an integration by parts, is 

g (e] = L f_/' (f) log sin2 

3-34. AbeVs theorem for power series. When for any fixed value of 
the Fourier series converges to a sum which may be denoted for the 
moment by g (6), it may be shown with the aid of a property of power 
series discovered by N. H. Abel that V -> g (6) as r -* 1. But since 

* Acta Math. vol. xxx, p. 335 (1906). 

t Crette's Journ. vol. oxu, p. 12 (1912). 

J Amer. Math. Soc. Colloquium Publication*, vol. vi (1927). 

16-2 



244 Two-dimensional Problems 

V -> F (0) we must have g (9) = F (0) and so the Fourier series represents 
F (9) whenever it is convergent. 

The series for V may be written in the form 

V = U Q + ru l + r 2 u z + ..., (G') 

and it should be noted that the coefficients r, r*, ... occurring in the different 
terms are all positive and form a decreasing sequence. The theorem to be 
proved is applicable to the more general series 

V = t>o^o + fli^i + v 2 u 2 + 
where the factors v , v l9 v 2 are all positive and such that 

Vu< ^ n , v = ! 
Let us write 

*0 = ^0> 1 = ^0 + %> S 2 = U + % + U 2 , 

and suppose that the quantities s , ^ , s 2 , ... possess an upper limit H and 
a lower limit A, then 

A % s n < //, for n = 0, 1, 2, .... 

Now if V n denotes the sum of the first n + \ terms of the series V, 
V - v Uo H- ViUi + ... ?; w ?/ n 

= Oo - '^l) ^0 + ( V l - V 2 ) ! + ... (V n _! ~ V n ) 5 n _! + V n S n , 

and in this series not one of the partial sums s m has a negative coefficient. 
Hence 

V n < K - v i) 7/ - f ( ?; i ~ v a) 7/ -H K-i - ^n) ^ + v n H> 
and K n > (?' - Vj) li + (i\ - v 2 ) h + ... (v n-1 - v w ) A + v w A. 

Summing the two series we obtain the inequality 

v A -r F n < v //, 

which shows that | V n \ < v n k, where h is a fixed quantity greater than 
either | h \ or | H \ . 
Similarly, if 

h n m V m U m + v m+l u m+l ~f~ v m+n u m+n> 

we have the inequality 

I 7? m I *f 11 Is 

I "'n I < v m K m> 

where k m is a positive quantity greater than any one of the quantities 

I u m | , | n m -f u m+l | , | u m + u m+1 + ^ m+2 | , .... 

If now the series U Q 4- u^ 4- w 2 + ^ s convergent and is any arbitrarily 
chosen small positive quantity, a number m (e) can be found such that 

| Um + WWH-I+ ... U m+n | < , 

for n ^ 0, 1, 2, ... and m > w (f). When m is chosen in this way we may 
take k m = and since v m < v < 1 we have the inequality 

I R n I < . 



AbeVs Convergence Theorem 245 

When the quantity v n is a function of a variable r which lies in the unit 
interval < r < 1 the foregoing inequality shows that the series (G') is 
uniformly convergent for all values of r in this interval and so represents 
a continuous function of r. In the case under consideration we have 
v n = r n and the conditions imposed on v n are satisfied if < r < 1. The 
function V is consequently continuous at r 1 and so 

U Q -f u^ + u 2 + ... = lim P (r, 0) = F (9). 



i 



3-41. The analytical character of a regular logarithmic potential*. 
Poisson's integral may be used to prove that a logarithmic potential V 
which is regular in a region D is an analytic function of x and y. 

We may, without loss of generality, take the origin at an arbitrary 
point within D. Let C denote the circle x 2 -f y 2 a 2 which lies entirely 
within D, then for a point x = a cos a, y a sin a on this circle, V = / (a), 
where / (a) is a continuous function of a and so by Poisson's formula 



V " J a 2 f r 2 - 2ar cos (0 - a) ' 



where x = r cos 9, y = r sin 9 and r < a. 
Now the series 



*.2 oo / r \ n 

, ,= 1 + 2 S (') cosn(0-) 



a 2 _|_ r 2 __ 2ar cos (9 a) tl ^i \a) 

is absolutely and uniformly convergent and so can be integrated term by 
term after being multiplied by / (a) rfa/2?r. Therefore 

F = a -f 2 f- ) (a n cos TI^ + b n sin ?i0). 

n-l \ a / 

Now if in the polynomial 

T\ 

- ) (a n cos ?i^ + 6 n sin n9) 

a/ 

= l a ~ n K ( + *y) n + a n (# - *V) W - *n (^ + %)" "I" ^n (# ~ ^) W ] 

we replace each term of type c, vq x *'tf l ^y ^ s modulus, the resulting expres- 
sion will be less than the corresponding expression obtained by doing the 
same thing to each term of type e PQ x p y q in the expansion of each of the 
four binomials and adding the results. Now this last expression is less than 



(T\ 
- ) 
a/ 



where M is the upper bound of a n and b n . Now let 

| x \ < ,9, | y | < s, 
where s < a/2, then 

2 [| x | + | y \] Ma~ n < 2 (2s/a) n M, 
and the series of moduli is convergent. The series for F is thus a power 

* E. Picard, Cours d* Analyse, t. n, p. 18. 



246 Two-dimensional Problems 

series in x and y which is absolutely convergent for | x \ < s, \ y \ < s, it 
thus represents an analytic function. 

Since, moreover, the origin was chosen at an arbitrary point in D it 
follows that V is analytic at each point within D. 

For the parabolic equation ~- = y^ there is a theorem given by 

Holmgren * which indicates that z is an analytic function of x in the neigh- 
bourhood of a point (x , y Q ) in a region R within which z is regular. 

If through the point (x , y Q ) there is a segment a < y < b of the line 
x = X Q which lies entirely within J?, there is a number c such that for 
| x X Q | < c, a < y < b there is an expansion 

//r _ T ^2n (rf __ 

z (x, y) = S eLL #<> (y) + 2 



where $ (y) = * (*o> y), $(y) = -fa z ( x o> V)- 

These functions < (?/), (y) are continuous (D, oo) in a < y < 6 and 
their derivatives satisfy inequalities of type 

| </> (n) (y) | < Mc~ 2n (2n) !, | (n) (y) | < Mc~* n (2n) ! . 



3-42. Harnack's theorem^. Let JF a , for each positive integral value of 
,9, be a potential function which is continuous (D, 2) (i.e. regular) in a closed 
region R and let the infinite series 

^i 4- w 2 + w 3 + ... ...... (A) 

converge uniformly on the boundary J5 of R, then the series converges 
uniformly throughout R and represents a potential function which is 
regular and analytic in R. The sum w n -f iv n+l + ... w n +v i g a potential func- 
tion regular in R. If it is not a constant it assumes its extreme value on B 
and if N p is the numerically greatest of these we shall have 

| w n + w n+l + ... w n+P I < \N 9 \. 

Since, however, the series converges uniformly on B we can choose a num- 
ber m (e) such that when n > m (c) we have 

\N,\<*. 

for all positive integral values of p. This inequality, combined with the 
previous one, proves that the series (A) converges uniformly in R and so 
represents a continuous function w. Now let C be any circle which lies 
entirely within 7? and let Poisson's formula be used to obtain expressions 
for potential functions W, W ly W 2 , W 3 , ... regular within C and having 
respectively the same boundary values as the functions W 9 w l9 w 2 ,w 3 , ... 

* E. Holmgren, ArJnv for Mat., Astr. och Fyaik, Bd. I (1904); Bd. m (1906); Bd. iv (1907); 
Comptes Rendus, t. CXLV, p. 1401 (1907). 

t Kellogg calls this Harnack's first theorem. See Potential Theory, p. 248. The theorem was 
given by Harnack in his book. It has been extended to other equations of elliptic type by 
L. Lichtenstein, Crelle'a Journal, Bd. CXLII, S. 1 (1913). 



Analytical Character of Potentials 247 

Since a potential function with assigned boundary values on C is unique 
if it is required also to be regular within C we have W s w s (s = 1, 2, ...). 
Furthermore, since the series (A) is uniformly convergent it may be inte- 
grated term by term after multiplication by the appropriate Poisson factor. 
Therefore at any point within 

w = w,+ w 2 + w 3 + ... 

= U\ ~\~ W 2 -f M> 3 + ... = W. 

Hence within C the function w is identical with the regular potential func- 
tion which has the same values as w at points on C. Since C is an arbitrary 
circle within R it follows that w is a regular potential function at all points 
of R and is consequently analytic at each point of R. 

For recent work relating to the analytical character of the solutions of 
elliptic partial differential equations reference may be made to L. Lichten- 
steiri, Enzyklo'padie der Math. Wiss., n C. 12 ; T. Rado, Math. Zeits. Bd. xxv, 
S. 514 (1926); S. Bernstein, ibid. Bd. xxvm, S. 330 (1928); H. Lewy, 
Gott. Nachr. (1927), Math. Ann. Bd. ci, S. 609 (1929). 

3-51. Scliwarz's alternating process. H. A. Schwarz* has used an alter- 
nating process, somewhat similar to that used by R. Murphy f in the treat- 
ment of the electrical problem of two conducting spheres, to solve the first 
boundary problem of potential theory for the case of a region bounded by 
a contour made up of a finite number of analytic arcs meeting at angles 
different from zero. 

To indicate the process we consider the simple case of two contours 
aa, 6/3 bounding two areas A, B which have a common part C bounded 
by a and /?, while a and 6 bound a region D represented by A f B C. 

We shall use the symbols a, b y a, /j to denote also the parameters by 
means of which the points on these curves may be expressed in a uniform 
continuous manner and shall use the symbols m and n to denote the points 
common to the curves a and b. We shall suppose, moreover, that the 
choice of parameters is made in such a way that in and n are represented 
by the parameters m and n whether they are regarded as points on a, 6, a 
or /?. This can always be done by subjecting parameters chosen for each 
curve to suitable linear transformations. 

Our problem now is to find a potential function V which is regular 
within D and which satisfies the boundary conditions V f (a) on a, 
V - g (b) on 6, where / (m) = g (m), f (n) = g (n). 

We shall suppose that / (a) is continuous on a and that g (b) is con- 
tinuous on b. We shall suppose also that a function h (a), which is con- 
tinuous on a, is chosen so as to satisfy the conditions 
h(m) =/(m), h (n) =/(/&). 

* Berlin MonatsberichU (1870); Gesammelte Werke, Bd. n, S. m 
t Electricity, p. 93, Cambridge (1833). 



248 Two-dimensional Problems 

We now form a sequence of logarithmic potentials u 1} u 2) ... regular in 
A, and a sequence of logarithmic potentials v l9 v 2 , ... regular in J3; these 
potentials being chosen so as to satisfy the following boundary conditions 
in which u s (/?) denotes the value of u s on /3, and v s (a) denotes the value of 

v s on a, (s ~ 1, 2, ...): 



on a, u^ ti (a) on a, 
u 2 = / (&) on a, i 2 = v (a) on a, 
u 3 ^ f (a) on a, ^ 3 = v 2 (a) on a, 

^ = gr (ft) on 6, #!_ = Uj (j8) on /3, 
^2 ~ CO on 6, v% u 2 (/3) on /3, 
# g -~ r/ (6) on 6, ?> 3 ?/ 3 (/3) on /3, 

Writing w s . f u L -f- (?/ 2 ?^j) |- (w 3 w 2 ) -h ... (u h u s ^i), 

our object now is to show that as .9 -> oo the series for u s and v s converge 
and represent potentials which are exactly the same in C. To establish the 
convergence of the series we shall make use of the following lemma. 

We note that w s = u s 11^^ is a logarithmic potential which is regular 
in A and which is zero on a. Let S,. (a) be its value at a point on a and let 
83 be the maximum value of | 8 S (a) \ . 

Now let </> be the logarithmic potential which is regular in A and which 
satisfies the boundary conditions (/ -= 1 on a, on a. As the point 
(x, y) approaches one of the points of discontinuity m, <f> tends to a value 
9 such that < 6 < 1. Now a regular potential function attains its greatest 
value in a region on the boundary of the region, therefore <f> < for all 
points of A and so there is a positive number e between and 1 such that, 
on /?, <f> < e < 1 . 

Now w s -f S s ,< is zero on a and positive on a and is a logarithmic 
potential regular in A . Its least value is therefore attained on the boundary 
of A and so w s f <.(/> > within A. This inequality may be written in the 

f rm o/i \ , . * n 

o s (<p j ~f- w\ -!- to, > 0, 

and since </> < e on /3 it follows that w s 4 e8 s > on j8. 

In a similar way we can show that iv s 8 s <f> < in A and so we may 
conclude that w s e8 s < on /?. Combining the inequalities we may write 

The number e was derived from the function <f> associated with A. In a 
similar way there is a number T? associated with the region B and the 
curve a. Let K be the greater of these two numbers if the two numbers are 
not equal. 



Schwarz*s Alternating Process 249 

Writing t s = v s ~ v s ^ and using the symbol r s (/3) to denote the value 
of t s on p we use r a to denote the maximum value of | r s (/3) | on /?. We then 
find in a similar way that , M , 

K I < T S , 

and so we may write 

| W a | < K8 S1 | t, < KT S . 

We thus obtain the successive inequalities 

I ' 2 ~ *'i I '-" I 
Therefore r 2 



\ on . 

<j - - j i 

Therefore S^ < /cr 2 < * 2 S 2 , ... S, +2 < * 2s S 2 , 

T 3 < *S 3 < K 2 T 2 , ... T s+2 < * 2 *T 2 . 

The series for ?/, and v s thus converge uniformly at all points of the 
boundary of C and so by Harnack's theorem represent regular logarithmic 
potentials which we may denote by ?/ and v respectively. Since ?/,, - v^^ 
on a and u s = v s on /3 it follows that u -= v on the boundary of 6" and so 
u = v throughout C. Since, moreover, the series for u converges uniformly 
on the boundary of A and the series for v converges uniformly on the 
boundary of B these series may be used to continue the potential function 
u v beyond the boundary of G into the regions A and B, and the potential 
function thus defined will have the desired values on a and 6. 

3-61. Flow round a circular cylinder. To illustrate the use of the com- 
plex potential in hydrodynamics we shall consider the flow represented by 
a complex potential % which is the sum of a number of terms 

y "* 

Xl - U (z + a 2 /z), X 2 = M log z, ^ = log ~ , 

z -- z t 

, i 2 -- 2 2 
X4 = -1C log 2 . 

^ ^3 

Writing z = re^, ^ = ^> 4- i^ we consider first the case in which x = Xi 
and 7 is real. We then have 

u~- iv= d x /dz = T7 ( 1 - a 2 /z 2 ) , 

^ = C7 sin (r - a 2 /^)- 

The stream-function </r is zero on the circle r 2 = a 2 and also on the line 
y = 0. There is thus one stream-line which divides into two parts at a 
point S where it meets the circle ; these two portions reunite at a second 
point S f on the circle and the stream -line leaves the circle along the 
line y = 0. Since z 2 = a 2 at the points S and 8' these points are points of 
stagnation (u = v = 0). It will be noticed that the stream-line y = cuts 
the boundary r 2 = a 2 orthogonally. This is in accordance with the general 
theorem of 1-72. 



250 Two-dimensional Problems 

At a great distance from the circle we have u iv = U, fy = C/y, and 
so the stream-lines are approximately straight lines parallel to the axis 
of x. Our function i/j is thus the stream -function for a type of steady flow 
past a circular cylinder. This flow is not actually possible in nature, the 
observed flow being more or less turbulent while for a certain range of 
speed depending upon the viscosity of the fluid and the size of the cylinder, 
eddies form behind the cylinder and escape downstream periodically* in 
such a way as to form a vortex street in which a vortex of one sign is 
almost equidistant from two successive vortices of the opposite sign and 
each vortex of this sign is almost equidistant from two successive vortices 
of the other sign. Vortices of one sign lie approximately on a line parallel 
to the axis of x and vortices of the other sign on a parallel line. 

Some light on the formation of this asymmetric arrangement of 
vortices is furnished by a study of the equilibrium and stability of a pair 
of vortices of opposite signs which happen to be present in the flow round 
the circular cylinder. 

The flow may be represented approximately by writing 

X = Xi + X* + *4> 

and choosing 2 , z l9 z 2 , z 3 so that the circle r 2 = a 2 is a stream -line, This 
condition may be satisfied by writing 



ItA,B,C,D are the points specified by the complex numbers z , z l , z 2 , z 3 , 
respectively, these equations mean that B is the inverse of A and the 
inverse of D. 

In the theory of Helmholtz and Kelvin vortices move with the fluid. 
When the vortices are isolated line vortices this result is generally replaced 
by the hypothesis that the velocity of any rectilinear vortex to is equal 
to the resultant of the velocities produced at its location by all the other 
vortices which together with o> produce the resultant flow at an arbitrary 
point. In using this hypothesis the uniform flow U is supposed to be 
produced by a double vortex at infinity and the complex potential Ua 2 /z 
is interpreted as that of a double vortex at the origin of co-ordinates 0. 

The vortex at A will be stationary when 



Taking for simplicity the cavse when r 2 = r , ft 2 = , c' = c, and 

* Th. v. lUrmAn, Odtt. Nachr. p. 547 (1912); PJiys. Zeits. p. 13 (1912). The vortices have 
been observed experimentally by Mallock, Proc. Roy. Soc. London, vol. ix, p. 262 (1907); and 
by Benard, Comptes Rendus, vol. CXLVJI, pp. 839-970 (1908); vol. CLVI, pp. 1003-1225 (1913); 
vol. OLXXXII, pp. 1375-1823 (1926); vol. CLXXXIII, pp. 20-184 (1920). 



Cylinder and Isolated Vortices 251 

separating the real and imaginary parts of the expression on the right, 
after multiplying it by 2 , we obtain the equations 

= U (r - a 2 r Q ~ l ) cos - c cot Q + a*cr Q 2 l- 1 sin 20 , 
= U (r + a 2 r -!) sin - cr 2 /(r 2 - a 2 ) - \c + cr 2 (r 2 - a 2 cos 20 ) Q-i, 
where 3 = r 4 - 2a 2 r 2 cos 20 + a 4 . 

The first equation gives 

2UQ, sin =- cr (a 2 - r 2 ), 
and when this value of U is substituted in the second equation it is found 

i/nat) 9 i ci 9 /i 

r 2 - a 2 = 2r 2 sm0 . 

This result was obtained by Foppl*, who also studied the stability of the 
vortices. The result tells us that the vortex can be in equilibrium if 
AB = AD. To confirm this result by geometrical reasoning we complete 
the parallelogram BADE and determine a point N on the axis of y such 
that ON = AN. Let M be the point of intersection of BC and AN , G the 
point of intersection of AC and BD. 

On the understanding that all lines used to represent velocities are to 
be turned through a right angle in the clockwise direction the velocities 
at A due to the different vortices may be represented as follows : 

Those due to the vortices at B and D by c/AB and c/AD respectively. 
Since AB = AD these two velocities together may be represented by 
c.AE/AB* along AE. 

The velocity due to the vortex at G may be represented by c/CA along 
GA and equally well by cGA/AB* along GA. The resultant velocity at A 
due to the vortices at B, C and D may thus be represented by c.GE/AB 2 
along OE. 

On the other hand, the velocity U is represented by U along ON, and 
the velocity due to the double vortex at O by U .NM/ON along NM. The 
velocity in the flow round the cylinder in the absence of the vortices is 
thus represented by U .OMJON. 

A A A 

Now MAE = ME A = OCM, therefore 0, M , A, C are concyclic and so 
0&LC = OAC - OEG. This means that OM and EG are parallel. By 
choosing c so that c.EG/AB 2 = U.OM/ON the resultant velocity at A 
will H zero. Since the triangles ON A, OAD are similar, the equation for 
c becomes simply 



T7 OM AB 2 T7 OM AB* TJ OM A n A ^ TT , 
c - U ON EG - U ON AG = U ON A = A C * ' U ' a 

- U (r 2 - a 2 ) (1 - a 4 /r 4 )/a 

and implies that the strength of the vortex at A is greater the greater the 
distance of A from the origin. 

* L. Foppl, Munchen Sitzungsber. (1913). See also Howland, Journ. Roy. Aeron. Soc. (1925); 
M. Dupont, La Technique Atronautique, Dec. 15 (1926) and Jan. 15 (1927); W. G. Bickley, Proc. 
Roy. Soc. Lond. A, vol. cxix, p. 146 (1928). 



252 Two-dimensional Problems 

The stream-lines in the flow studied by Foppl are quite interesting and 
have been carefully drawn by W. Miiller*. There are four points of 
stagnation on the circle, two of these, 8 and 8' ', lie on the line y = 0, while 
the other two, S , $</, are images of each other in the line y = 0. Stream- 
lines orthogonal to the circle start at S Q and S ' and unite at a point T on 
the line y where they cut this line orthogonally. This point T is also a 
point of stagnation. Outside these stream -lines the flow is very similar 
to that round a contour formed from arcs of two circles which cut one 
another orthogonally; within the region bounded by these stream-lines 
there is a circulation of fluid and the flow between T and the circle is 
opposite in direction to that of the main stream. The stream-lines are, 
indeed, very similar to those which have been frequently observed or 
photographed in the case of the slow motion round a cylinderf. 

Let us now consider the case when there is only one vortex outside the 
cylinder and a circulation round the cylinder. We now put 

A Al ' A2 ' A3 * 

In this case 

u - w ={7(1- a*/z 2 ) -f ik/z -f ic [(z - r Q e td )~ l - (z - v^aVo)- 1 ], 
and the component velocities of the vortex A are given by 

while for its image B 

?/! -f ii\ -= a 2 (U Q iv Q ) r ~ 2 e 2l V 

If X, Y are the components of the resultant force on the cylinder per 
unit length, we have 

X + iY --= - Jpa P* (w 2 + v 2 -f- 2 -} e"dO - (X g + iY g ) -I- (X* + iY+), say. 
J o V v* / 

Now when z ae t0 9 
u" + v 2 = 4C/ 2 sin 2 8 + k*fa 2 4- c 2 (r 2 - a 2 ) 2 /a 2 7? 4 

-I- 4C7 sin 0.[*/a - c (r 2 - a 2 )/a.R 2 J - 2kc (r 2 - a 2 )/a 2 J? 2 , 
where 7^ 2 = a 2 + r 2 - 2ar cos (/9 - ). 

Therefore ^ + iY q = 2-rrip {kU c (U Q -f- iv )}. 

We have also for r a 

* Znts.fur techmsche Physik, Bd. vm, S. 62 (1927); Mathematische Stromungslehre (Springer, 
Berlin, 1928), p. 124. 

t See especially the photographs published by Camichei in La Technique A fronautique, Nov. 15 
(1925) and Dec. 15(1925). 



Circulation round a Cylinder and Vortex 253 

therefore 

\*'W , . .-9f| 



n 



* a/ 

a< . 9</r 

a* + * dt 

2rr gj, 

- ae t<? <i# == Trie (Uj + ivj) ~ 7rica 2 r Q ~ 2 (U Q iv ) e 2 * e o 2Tric (t^ -f ivj). 
o ot 

Combining these results we have 

X + iY = 27rip {/J?7 c (w + i# ) -f c (^ + i#i)}. 

This result may be extended to the case in which there are any number 
of vortices outside the cylinder*, the general result being 

-X" -f iY = 2 Trip \kU 2 c s (?/ 2s 4- iv 2s 

In the special case when there is only one vortex and k = 0, = 0, 
we have u t -f iz^ = a 2 r Q ~ 2 (U Q ~ iv Q ), 

, MJ TT /I n 2 r 2\ ^> r /r 2 /t2\-l 

U-Q frt/Q \J ^1 U/ f Q f t/Ll Q \' Q ^ / > 

Z f iF = 27rpc [c/r Q iU (I ~ a 4 r ~ 4 )]. 

Introducing the coefficients of lift and drag, defined by X ~ p8U 2 .C J)9 
Y = pSU 2 .C L , 8 being the projected area, we find 

C D = (c/aU) 2 7ra/r , C L = - TT (c/aC7) (1 - a*r - 4 ). 

These results were obtained by W. G. Bickleyf who plots the lift-drag 
curves for r - 2a, 4a and 6a, and compares them with the published 
curves for Flettner rotors (rotating cylinders with end plates). With the 
last two values the agreement is fair except for low values of the lift. 

The stream-lines for the case of a single vortex outside the cylinder 
have been drawn by W. MiillerJ. 

EXAMPLES 

1. If in a type of flow similar to that considered by Foppl the vortices at 2 and z 2 are not 
images of each other in the line y = 0, one of the conditions that the vortices may be stationary 
in the flow round the cylinder is 

(r - a 2 r Q ~ l ) cos - (r 2 - a*r 2 ~ l ) cos a . 

2. If in Foppl's flow the vortices move so that they are always images of each other in 
the line y the resultant force on the cylinder is a drag if 

4r * sin 2 > (r 2 - a 2 ) 2 . 

[Bickley.] 

* H. Bateraan, Bull. Amer. Math. Soc. vol. xxv, p. 358 (1919); D. Riabouchinsky, Comples 
Rendus, t. CLXXV, p. 442 (1922); M. Lagally, Zeits. f. angew. Math. u. Mech. Bd. n, 8. 409 (1922). 
In this formula the even suffixes refer to the vortices outside the cylinder and the odd suffixes 
to the image vortices inside the cylinder. 

t Loc. cit. ante, p. 251. 

t Loc. cit. ante, p. 252. 



254 Two-dimensional Problems 

3. A plate of width 2a is placed normal to a steady stream of velocity U and vortices 
form behind the plate at the points 

Prove that the conditions are satisfied by 



= *. 

8 (* + *)* -(* + ')* 
Prove also that when 



- * + 2 ( V + a 2 )*, 

the velocity does not take infinite values at the edges of the plate and the vortices are 
stationary. 

[D. Riabouchinsky.] 

3-71. Elliptic co-ordinates. Problems relating to an ellipse or an 
elliptic cylinder may be conveniently solved with the aicf of the sub- 

stitution . . U /<" , \ 1 Y 

x + ly = c cosh (f + 2,7?) = c cosh f , 

which gives 

x = c cosh cos 77, 

T/ = c sinh sin 77. 
The curves = constant are confocal ellipses, 

5 2 + j__ = l 
c 2 cosh 2 f c 2 sinh 2 5 

the semi-axes of the typical ellipse being a = c cosh ^ and 6 = c sinh . 
The angle 77 can be regarded as the excentric angle of a point on the 
ellipse. 

The curves 77 = constant are confocal hyperbolas, the semi-axes of the 
typical hyperbola being a' c cos 77 and 6' = c sin 77. 

The first problem we shall consider is that of the determination of the 
viscous drag on a long elliptic cylinder which moves parallel to its length 
through the fluid in a wide tube whose internal surface is a confocal 
elliptic cylinder*. 

Considering a cylindrical element of fluid bounded by planes parallel 
to the plane of xy and a curved surface generated by lines perpendicular 
to this plane, the viscous drag per unit length on the curved surface of the 
cylinder is 



taken round the contour of the cross-section, w being the velocity parallel 
to a generator and p being the coefficient of viscosity. 

If the fluid is not being forced through the tube under pressure the 
pressure may be assumed to be constant along the tube and so in steady. 
motion the total viscous drag on the cylindrical element must be zero. 

* C. H. Lees, Proc. Roy. Soc. A, vol. xcn, p. 144 (1916). 



Viscous Resistance to Towing 255 

Transforming the line integral into an integral over the enclosed area, we 

obtain the equation 

d*w Shu 

fo 2 + ay 2 ~ ' 

The boundary conditions are w = when = ^ , and w = v when 
= 2 we therefore write 



Since T? varies from to 2-rr in a complete circuit round the contour of 
the cross-section, the total viscous force per unit length of the cylinder is 

27TfJLV __ 2-JT^JLV 

S^ =r ^2 ~ log (aT+ b~] T~Tog r (a 2 "+ 6 2 ) ' 

If the inner ellipse reduces to a straight line of length 2c, the total drag 
on the plane is D per unit length, where 

D [log (a, + 6,) - log (2c)] = 2^, 
and the resistance per unit area at the point x is 

(D/27r) (c 2 - z 2 )-*. 

It is clear from this expression that the resistance per unit area, i.e. the 
shearing stress, is much greater near the edges of the strip than near its 
centre line. 

The foregoing analysis may be used with a slight modification to 
determine the natural charges on two confocal elliptic cylinders regarded 
as conductors at different potentials. If V is the potential at (f, 77) and 
V = for = x , V = v for f = 2 , we have 

y & - &) = (& - a 

and the density of charge on the cylinder = ^ is 

i aF3 i 



_ 

~ 477 8 3/1 -~ 47T 87 35 ^ ST(^ - &) c 

When the inner cylinder reduces to the strip whose cross-section is 
S^^ we have, when v = 2 ( x ^ 2 ), 

CTj = (1/277-C) COS6C T^u 

and if, moreover, the outer cylinder is of infinite size o l becomes the natural 
charge on the strip when the total charge per unit length is equal to 
unity; this is the charge density on each side of the strip. 

To find the stream-function for steady irrotational flow round an 
elliptic cylinder when there is no circulation round the cylinder, we write 
^ = ^i -f- 02 > where 

^i ^ Uy ~ Vx = c (U sinh ^ sin 77 V cosh cos 77) 
is the stream-function. for the steady flow at a great distance from the 



256 Two-dimensional Problems 

cylinder and ^ 2 is the stream-function for a superposed disturbance in this 
flow produced by the cylinder. To satisfy the boundary condition </> = 
at the surface of the cylinder, and the condition that the component 
velocities derived from </f 2 are negligible at infinity, we write 

2 = e~* (A cos ?; -f B sin ??). 

Choosing the constants so that tft on the cylinder, we have 
e-fi A = cV cosh & , e~ f i B = cU sinh & 

= ^7 =-6 t Z7 

where a l5 6 X are the semi-axes of the ellipse = &. We have also 

!-{-&! = c^-^i, a x fej_ = Cjg-fi 
Therefore 2 = (a x + 6^* (% 6^"* e~ f (7% cos T? f/fij sin TJ), 

<f> + i*fi= (U - iV) z -f (*//>! + iFj) (aj + 6^4 (% - &!)-* e-f. 

To find the electrical potential of a conducting elliptic cylinder which 
is under the influence of a line charge parallel to its generators, we need an 
expression for the logarithm 

log (z - z) =-- log [c (cosh - cosh }] 

- -{- log Jc + log (1 - e^-^o) (l - e~^-^o) 

- & + logic -2 S w^e- 
Writing this equal to (/> -f ifa we have 

00 

^o "-^ ^o -H lg ^ 2 S n- 1 e~ nf o (cosh nf cos nrj cos 
>i-i 

+ sinh ng sin ?^r; v^in nrj Q ). 

To obtain a potential which is constant over the elliptic cylinder 
f ^ f i > wo write </> == ^ 4- ^ , where 

00 

^> 1 (^rl n e~" cos Tir; -f B n e~^ sin w^). 

71-1 

Each term of this series is indeed a potential function which vanishes 
at infinity. Choosing the constants A n , B n , so that the boundary condition 
< --= on = ^ is satisfied by <f> = <f> + <f> l9 we have 
nA n e~ n %i = 2e~ n ^o cosh T^ cos TIT^Q, 
nB n e~ n ti = 2e~ n ^o sinh n^ sin n^. 
Hence when ^ < ^ < , 



^ ~ f o + ^g l c + 2 7i- 1 e n( ^i"^o> sinh TZ- (^ g) 

n~ I 

Summing the series we find that 

.-.-J eL 



Induced Charge Density 257 

The corresponding stream-function is 



* - , + tan- -_ IL .+ tan- 

r ' 1 e*- f o cos (770 ??) 

and when ^ = the value of for = is 



__ _ __ 

cosh - cos (770 - 77) ' 
The surface density of the charge on the plate = is thus 

1 dr i [i _ ___ sinh/p 1 

4:77 ds L cosh cos (770 77)] ' 

and the total charge is zero. When the total charge per unit length is 1, 
and the total charge per unit length of the line is 1, the surface density 
of the charge on the cylinder is 

1 drj sinh 

, 27T ds cosh cos (779 77) " 

This is what C. Neumann* calls the induced charge density or the 
induced loading; it represents, of course, the charge on one side of the 
plate = 0. We shall write this expression in the form 

<r(0, 77; )>??o) ^^TT^fe /S ^' 7?; ^' 7?0 ^ 
and shall use a corresponding expression 

<*(fi>ih; o?7o) = 27T ~ ds s (i>*?i; o> ^o)* 

in which 

<* (t ' \ sinh (|o - ^) . 

s (&, ih, 6. %) = cosh ^--gy-.-ooa & - ^)' ...... (A) 

and or (fj, 77!; ^ , 770) is the density of the induced charge for the elliptic 
cylinder f = ^ . 

Let us now consider the problem in which a function V is required to 
satisfy the condition V = / (77!) on the cylinder = &, while V is a regular 
potential function outside the cylinder = ^ but not necessarily vanishing 
at infinity. Some idea of the nature of the solution may be obtained by 
first considering the two cases 



Since 



^ cos mr }9 V ~ e~<-i> cos W77, 
~ sin 77177, V e-"* ( -i> sin 77177. 



(&, %; f , 1?) - 1 + 2 S e-i> cos m (77 - 77,), 

w-l 

* Leipzig. JBer. Bd. LXH, S. 87 (1910). 

17 



258 Two-dimensional Problems 

the solution is given in these cases by the formula 

V = -S (&, %; & itf/fa) Ah ...... (B) 



and we may write 

o- (&, *h; , i?) = o- (1, i?i) 5 (&, ^; f, 77), ...... (C) 

where o > (^ 1 , 77^ is the natural density per unit length when the total 
charge per unit length on the cylinder f = ^ is unity. This is a particular 
case of a general theorem due to C. Neumann*, which tells us that the 
density of the induced charge for a cylinder whose cross-section is a closed 
curve can be found when the natural density on- the cylinder and the 
corresponding potential is known. The expression for the induced charge 
is then of the form (C), where and 77 are conjugate functions such 
that f = constant are the equipotentials and 77 = constant, the lines of 
force associated with the natural charge. The undetermined constant 
factor occurring in the expressions for functions and 77 which satisfy the 
last condition should be chosen so that 77 increases by 2?r in one circuit 
round the cross-section of the cylinder. 

The formula (A) gives a potential which satisfies the conditions of 
the problem for a wide class of functions and for this class of functions 
we have the interesting relation 

27r f M - Urn f ^ Sinh ( ^ ~~ &)/ foi) *h (f^f\ 
2 w /(,) - hm J o cogh (f _ ^ _ cog (i -j (f > ). 

The question naturally arises whether the function F given by (B) 
is the only function which fulfils the conditions of the problem. To discuss 
this question we shall consider the case when the ellipse f = f x reduces 
to the line = 0, i.e. the line S 1 S 2 . 

It will be noticed that when f (rj) = 1 the formula (B) gives 7=1. 
Now the potential <f> which is the real part of the expression 

(f> + i*ft=z (z 2 - c 2 )~i - coth , 

satisfies the condition that </> = on the line = and </> = 1 at infinity. 
Furthermore, the function <f> lt which is the real part of 

<i -f i^i = c (z 2 c 2 )"^ = cosech , 
satisfies the conditions 

</! = when = 0, fa = when = oo. 

Hence a more general potential which satisfies the same conditions 

as v is TT j i -.-^ * 

F+ A<f> + B<f> l9 

* Leipzig. Ber. Bd. LXII, S. 278 (1910). 



Munk^s Theory of Thin Aerofoils 259 

where A and B are arbitrary constants. Now 

sinh ^ 1 4- e-* +t *o 

cosh cos (T? 770) ~~ 1 e-+ <7 >o 

__ o sinh $ -f i sin T/ O 
cosh cos 170 * 
Hence, if 

* fsinh -f i sin 770 cosh A 



If* f 
- 

2 TT J _ [ 



- , - , -- -T 

2 TT J _ [ cosh - cos 770 smh 

...... (D) 

where?/ and F are constants, the potentials u and v are conjugate functions 
which can be regarded as component velocities in a two-dimensional flow 
of an incompressible in viscid fluid. These component velocities satisfy the 
conditions 

u U, v = F at infinity, v / (77) on the line $ x $ 2 . 

This result is of some interest in connection with Munk's theory of 
thin aerofoils. In this theory an element ds of a thin aerofoil in a steady 
stream of velocity U parallel to Ox is supposed to deflect the air so as to 

give it a small component velocity v = u -~ in a direction parallel to the 

Ci XQ 

axis of y. Assuming that u = U -f- c, where e is a small quantity of the 
same order of smallness as y Q and dy /dx , we neglect e , - , as it is of order 

CiXQ 

e 2 , and write v = U -~- . This is now taken to be the ^-component of 

(LXq 

velocity at points of the line S 1 S 2 and the corresponding component 
velocities (u, v) for the region outside the aerofoil may be supposed, with 
a sufficient approximation, to be given by an expression of type (D). 
In this expression, however, the coefficient A is given the value 1 so that 
the velocity at the trailing edge will not be infinite. 
Now when | | is large we may write 

(cosh - cos Tjo)- 1 = sech -f- cos TJ Q sech 2 + 

sinh = cosh | sech J sech 3 -f- ... , 
cosech = sech -f J sech 3 -f ____ 
Hence, when V = the flow at a great distance from the origin is 

f tyPe V+iu= iU + ft/* + &2 2 4 ... , 

c f ir 
where fa = 2 ^ j ^ ( 1 -f cos 770 -f i sin rj Q ) f fa) drj Q , 

c 2 ( n 
& = 2 - J _ (* sin 770 cos 770 - sin 2 770) / (770) <fy , 

17-3 



260 Two-dimensional Problems 

and by Kutta's theorem the lift, drag and moment per unit length of the 
aerofoil are given by the expressions of 4- 71 

L + iD = \p [(v+ iu) 2 dz^ 

M = \pR f (v -f w) 2 zdz = - 
Therefore L - - P cU 2 f* (1 + cos T? O ) J? y cfy , 

J - 7T ^#0 

sin =0, 



These are the expressions obtained by Munk* by a slightly different 
form of analysis. A more satisfactory theory of thin aerofoils in which the 
thickness is taken into consideration, has been given by Jeffreys and is 
sketched in 4-73. 

Since dx Q = c sin f] Q drj Q , X Q = c cos T? O , 



we may write L - - 2pU 2 [ (c + X Q ) (c 2 - x^ -^ dx Q 

J -c uX 

c 

= 2pcU 2 \ (c + x )-i(c-a; Hy ^ , 

J ~C 

M = 2pU 2 I (c 2 - ^ 2 )-* xyodx f . 

J -c 

3-81. Bipolar co-ordinates. Problems relating to two circles which 
intersect at two points S 1 and $ 2 with rectangular co-ordinates (c, 0), 
( c, 0) revspectively may be treated with the aid of the conformal 

transformation . . ., . . v 

z = ic cot |^, ...... (A) 

where z ^ x + iy, = 4- ^ and (^, t/), (f , 7^) are the rectangular co- 
ordinates of two corresponding points P and TT. We shall say that the 
point P is in the z-plane and the point TT in the -plane. The transformation 
may be said to map one plane on the other. 
It is easily seen that 



z - c 



where 



= ( x _ c) 2 -f s/ 2 - I z - c j 2 - 2cMe~\ 
= (a: + c) 2 -f 7/ 2 = | z + c | 2 



= -... 

cosh 77 cos f 

* National Advisory Committee for Aeronautics, Report, p. 191 (1924); see also J. S. Ames, 
Report, p. 213 (1925) and C. A. Shook, Amer. Journ. Math. vol. XLvm, p. 183 (1926). 



Bipolar Co-ordinates 261 

The curves = constant are clearly circles through the points 8 l and 
S 2 , while the curves 77 = constant p 

are circles having these points as 
inverse points. The two sets of 
curves form in fact two orthogonal 
systems of circles, as is to be ex- 
pected since the transformation (A) 
is conformal, and the corresponding 
sets of lines are perpendicular. S 2 

The expressions for x and y in Fi s- 1() - 

terms of f and rj are x = M sinh 77, y = M sin f . At a point P of the 
line ^$2 we have = TT, therefore 

# - c tanh 




and the natural loading for this line is 

o- = (I/we) cosh (iyo/2 ) 

The loading induced by a charge 1 at the point P (f , r?) is, on the 
other hand, 



_ 

7T0 [cosh (77 7? ) -f cos ] 

cosh fo/2) + cosh [fo/2) - 770] f 

CTn | , > ,. UUO rt 

cosh (Y] T? O ) -f cos f 2 

EXAMPLE 

A potential function v is regular in the semicircle y > 0, x* -f y 2 < a 2 and satisfies the 
boundary conditions v = A when t/ =- 0, ^ -= .B when x 2 + y 2 a 2 , prove that 



/(JO 

v = A-A' 
Jo 



( TT) dm 

m , O WTT 

cosh- 

where x -f t/y = ia cot j ( -f *), .4' = a#. 

3-82. Effect oj a mound or ditch on the electric potential. Let us now 
consider the complex potential 

2c 

where K is a real constant at our disposal. The potential <f> is zero when 
= 0, for in this case ^ becomes cot , and is a purely imaginary 

K. /C 

quantity. It is also zero when = |/CTT, for then # = tan ^ , and is 

again a purely imaginary quantity. The potential < is thus zero on a 
continuous line made up ,of the portion of the line y = outside the 
segment S 1 S 2 and of the circular arc through S 1 S 2 at points of which 



262 Two-dimensional Problems 

S 1 S 2 subtends the angle \KTT. Thus the complex potential x provides us 
with the solution of an electrical problem relating to a conductor in the 
form of an infinite plane sheet with a circular mound or ditch running 
across it. 

<,. d(f> (cosh r) cos f ) 2 dc/> 

dy cosh 77 cos 1 9f ' 

we find that on the axis of y, where 77 = 0, 



Al fy 2C of 

Also - = - - cosec 2 -, 

d /c 2 AC 



consequently the potential gradient on the axis x = is 

2 

2 cosec 2 - (1 cos |). 

At the vertex where = |KTT it is 

2 (1 cos I/CTT). 

On the plane y = 0, we have f - 0, and the gradient is 
cosech 2 -.(cosh -n I). 

K* K 

As ?? -vQ, x->co and the gradient tends to the value 1 which will be 
regarded as the normal value. 

As 77 -> oo, x -> c and the gradient tends to become zero or infinite 
according as K ^ 2. 





Fig. 17. Fig. 18. 

When K 1 we have a semicircular mound. 

The gradient on the line x = is everywhere greater than the normal 
value, at the vertex it is 2, and at a point at distance 2c above the vertex 
it is 10/9. 

When AC 3 we have a semicircular ditch. 

The gradient on the line x = is everywhere less than the normal 
value, at the bottom of the ditch it is 2/9 and at a point (0, c), at distance 
2c above the bottom, it is 8/9. By making K -> we obtain values of the 



. CL7T 

v cosec 2 



Electrical Effects of Peaks and Pits 263 

gradient for the case of a cylinder standing on an infinite plane. We must 
naturally make c -* at the same time, in order to obtain a cylinder of 
finite radius a. The appropriate complex potential is 

X = <f> + i$ = an cot --- . . ...... (C) 

On the line x = the potential gradient is 

CL7T\ 

, 

yr 

and tends to the normal value as y -> oo. 

When T/ == 2a the gradient is , which is nearly 2-5. At a distance 2a 

2 

above the summit, y = 4a and the gradient is ^ = 1-2337. On the axis 

8 

of x the gradient is 

a7T i**f a7r \ 

cosech 2 

x 2 \ x J 

As x -> the gradient diminishes rapidly to zero, consequently the 
surface density of electricity is very small in the neighbourhood of the 
point of contact. 

EXAMPLE 

Fluid of constant density moves above the infinite plane y = with uniform velocity U. 
A cylinder of radius a is placed in contact with the plane with its generators perpendicular 
to the flow. Prove that the stream function is derived from a complex potential of type (C) 
multiplied by U and calculate the upward thrust on the cylinder. 

[H. Jeffreys, Proc. Camb. Phil. Soc. vol. xxv, p. 272 (1929).] 

3*83. The effect of a vertical wall on the electric potential. Let h be the 
height of the wall, x = </> + ty the complex potential. If a is a constant, 

the substitution ., i m 

az = ih (a 2 + x) ...... (**) 

makes the point z = ih correspond to x = 0> the points on the axis of x 
correspond to the points on </> = for which 2 > a 2 , while the points on 
the axis of y for which y < h correspond to the points on <f) = for 
which 2 < a 2 . Hence, if </> be regarded as the electric potential, a con- 
ducting surface consisting of the plane y = and the conducting wall 
(x = 0, y < h) will be at zero potential*. 

If (r, 0), (/, 6') are the bipolar co-ordinates of a point P relative to $, 
the top of the wall, and to AS', the image of this point in the plane y = 0, 
we have 

h* x 2 = - a 2 (z 2 + h*)= - a 2 rr'e i(e+e '>. 
Therefore h<f> = a (rr r $ sin J (6 4- 0')> 

Jufi - - a (rr')i cos (0 + (9'). 

Therefore 2Afy 2 - a 2 {[(x 2 - i/ 2 + A 2 ) 2 + 4x 2 y 2 ]* - (x 2 - y 2 H- A 2 )}. 
* G. H. Lees, Proc. Roy. Soc. London, A, vol. xci, p. 440 (1915), 



264 Two-dimensional Problems 

The equipotentials have been drawn by Lees from the equation 

y 2 (1 + a 2 x 2 /h 2 cf> 2 ) = h 2 + x 2 -f h 2 <f> 2 /a*. 

To determine the surface density of electricity we differentiate 
equation (D), then 



When a: 0, z = it/, /^ = ia (A 2 y 2 )^, and so 



The* surface density is thus zero at the base and infinite at the top of 
the wall. 

When y = 0, z = x, /># = ia (h 2 -f x 2 )^, and so 



As a; -> oo this tends to the value a/h which may be regarded as the 
normal value of the gradient. At a distance from the foot equal to h the 
vertical gradient is 0-707 times the normal gradient. 

The curve along which the electric field strength has the constant value 
F is -given by 

a 2 (x 2 + y 2 ) - h 2 F 2 [(x 2 - y 2 + h 2 ) 2 + 

that i8 ' 



where (7?, 0) are the polar co-ordinates of P with respect to the origin. 
The curves F = constant may be obtained by inversion from the family 
of Cassinian ovals with 8 and $' as poles, they are the equipotential curves 
for two unit line charges at S and $', and a line charge of strength 2 at 
the origin 0. The rectangular hyperbola 

7/ 2 - X 2 - |/* 2 

is a particular curve of the family. This hyperbola meets the axis of y at 
a point where the horizontal gradient is equal to the normal gradient. 
The force is equal in magnitude to the normal gradient at all points of 
this hyperbola. At points above the hyperbola the force is greater than 
a/h, at points below the hyperbola it is less than a/h. 

The curves along which the force has a fixed direction are the lemnis- 
cates defined by the equation 

+ 0' 20 = constant. 

Each lemniscate passes through, S and S' and has a double point at 0. 
It should be noticed that the transformation 

az - ih (a 2 - & 
enables us to map the upper half of the -plane on the region of the upper 



Effect of a Vertical Wall 265 

z-plane bounded by the line y = and the vertical wall x = 0, y < h. This 
transformation makes the points at infinity in the two planes corre- 
sponding points. It may be observed also that if a 2 2 r 2 e 2ie , the 
angle 20 ranges from TT to TT. Hence, since az = hr sin 6 -f ihr cos 0, 
hr cos is never negative and so it is the upper portion of the cut z-plane 
Wjhich corresponds to the upper portion of the -plane. If we invert the 
z-plane from a point on the negative portion of the axis of y we obtain 
a region inside a circle which is cut along a radius from a point on the 
circumference to a point not on the circumference. The upper half of the 
-plane maps into the interior of this region. 

If, on the other hand, we invert from the origin of the z-plane, the 
cut upper half plane inverts into a half plane with a cut along the y-axis 
from infinity to a point some distance above the origin. The point at 
infinity in the -plane now maps into the origin in the z-plane. 



CHAPTER IV 

CONFORMAL REPRESENTATION 

4 11. Many potential problems in two dimensions may be solved 
with the aid of a transformation of co-ordinates which leaves V 2 7 = 
unaltered in form. It is easily seen that the transformation 

= / (*> y)> i? = 9 ( x > y) 

furnished by the equation 

l=g + ii, = F(x + iy) = F (z) 

possesses this property when the function F is analytic, because a function 
of which is analytic in some region F of the -plane is also analytic in 
the corresponding region G of the z-plane when regarded as a function of z. 
In using a transformation of this kind it is necessary, however, to be 
cautious because singularities of a potential function may be introduced 
by the transformation, and the transformation may not always be one-to- 
one, i.e. a point P in the -plane may not always correspond to a single 
point Q in the z-plane and vice versa. 

Let F be a function of f and 77 which is continuous (D, 1), then 

37^373 aFcfy 37^3731 dVdrj 
dx 3 dx drj 3x J dy dg dy 877 dy' 

These equations show that if the derivatives of. and 77 are not all 
finite at a point (x, y) in the z-plane, the derivatives ~ , ~ may be infinite 

37 dV 

even though -^ and _ are finite. A possible exception occurs when 

3F 37 

yr and -x both vanish, i.e. at a point of equilibrium or stagnation. 

At any point (# , y Q ) in the neighbourhood of which the function F (z) 
can be expanded in a Taylor series which converges for | z z | < c, 
we have 

F (z) = 1 a n (z - z ), 

n-O 

where z = x + iy, z = X Q + iy , 

and if = f + i, = J (z), = + irj Q = F (z ), 

we may write 

d - - = F (z) - ^ (z ) = rfz [f (z) + e], 
where rfz = z z and e -> as dz -> 0. 

Hence d = dz.^ 1 ' (x) approximately. 



Properties of the Mapping Function 267 

This relation shows that the (x, y) plane is mapped conformally on the 
(f , rj) plane for all points at which | F' (z) \ is neither zero nor infinite. 
We have in fact the approximate relations 

da = ds | F' (z) | , 
<f>= 6+ a, 
where dz = ds.e ie , dt, = dv.e^, 

F' (z) = | F' (z) | 4*. 

These relations show that the ratio of the lengths of two corresponding 
linear elements is independent of the direction of either and that the angle 
between two linear elements dz, 8z at the point (x, y) is equal to the angle 
between the two corresponding linear elements at (, 77). The first angle is, 
in fact, 6 6' ', while the second angle is 

<!>-<l>'=(0 + a)-(8' + a) = 0- 6'. 

These theorems break down if some of the first coefficients in the 
expansion 

- ^^^a n (z~z.Y (A) 

n-l 

are zero. If, for instance, a^ = a 2 = ... a m ^ = 0, we have for small values 

Of | Z - ZQ | 

- o = a m (* ~ Zo) m > 
and the relation between the angles is 

, <f> = m0 + a m , where a m = \a m \ e"m. 
This gives 

<f> - </>' - m (0 - 6'). 

More generally if there is an expansion of type (A) in which the 
lowest index m is not an integer a similar relation holds. 

4- 12. The way in which conformal representation may be used to 
solve electrical and hydrodynamical problems is best illustrated by means 
of examples. One point to be noticed is that frequently the transformation 
does not alter the essential physical character of the problem because an 
electric charge concentrated at a point (line charge) corresponds to an 
equal electric charge concentrated at the corresponding point, a point 
source in a two-dimensional hydrodynamical problem corresponds to a 
point source and so on. 

These results follow at once from the fact that if <f> + i$ is the complex 
potential we may write 

<t> + i*f>=f(x+ iy) - g (f + iy), 

and if <f> is the electric potential, the integral \difj taken round a closed curve 
is 47T times the total charge within the curve. Now the interior of a 



268 Confonnal Representation 

closed curve is generally mapped into the interior of a corresponding closed 
curve and a simple circuit generally corresponds to a simple circuit, 
moreover is the same in both cases and so the theorem is easily proved. 
It should be noted that a simple circuit may fail to correspond to a simple 
circuit when the closed curve contains a point at which the conformal 
character of the transformation breaks down. Another apparent exception 
arises when a point (x, y) corresponds to points at infinity in the (, rj) 
plane, but there is no great difficulty if these points at infinity are imagined 
to possess a certain unity. In fact mathematicians are accustomed to 
speak of the point at infinity when discussing problems of conformal 
representation. This convention is at once suggested by the results obtained 
by inversion and is found to be very useful. There is no ambiguity then 
in talking of a point charge or source at infinity. 

We have sen that certain angles are unaltered by a conformal trans- 
formation and can consequently be regarded as invariants of the trans- 
formation. Certain other quantities are easily seen to be invariants. 
Writing 



where d (x, ?/), d (, 77) are elements of area in the two planes, we have 



J ( _L - r \ - ^ ^ _i_ 

W '"dr"' 1 " 



J 



Sx 



The quantities </> and are usually taken to be invariants in a con- 
formal transformation and the foregoing relations indicate that 



* ffl"V9<A\ 2 /3^\ 2 1 7 ^ 

. and \\\(J} +U \dxdy 

cx 2 y 2 / }}[\dxj \dyj J * 

are invariants. In the theory of electricity the first integral is proportional 
to the total charge associated with the area over which the integration 
takes place. In hydrodynamics the second integral represents the total 
vorticity associated with the area and the third integral is proportional 
to the kinetic energy when the density of the fluid is constant. The 

invariant character of the integrals (udx -f vdy) and (udy vdx) is easily 
recognised because these represent \d(f> and \difs respectively. 



Riemann Surfaces and Winding Points 269 

4*21. The transformation w = z n . When n is a positive integer the 
transformation w = z n does not give a (1, 1) correspondence between the 
w-plane and the z-plane but it is convenient to consider an ti-sheeted 
surface instead of a single plane as the domain of w. For a given value of 
w the equation z n w has n roots. If one of these is Z l the others are 
respectively Z 2 = Z^, Z 3 = Z^ 2 , ... Z n = 2 1 o> w - 1 , where oj = e 2rrt / n . 

If 2 = re ie we may adopt the convention that for 

Z ly 0< 710 < 2,T, 
Z 2 , 2n< n9< 477, 



Z n , (2W 2) 77 < 710 < 2/177. 

Defining the sheet (m) to be that for which W m = Z m n we can say that 
W is in the first sheet, PF 2 in the second sheet, and so on. The n sheets 
together form a "Riemann surface" arid we can say that there is a (1, 1) 
correspondence between the z-plane and the Riemann surface composed 
of the sheets (1), (2), ... (n). If w --= Re 1 we have = n9, and so when 
w = W^n we have (2m 2) 77 < < 2w7r. 

The z-plane is .divided into n parts by the lines joining the origin to 
the corners of a regular polygon, one of whose corners is on the axis of x. 
These n portions of the z-plarie are in a (1, 1) correspondence with the 
n sheets of the Riemann surface. The n lines just mentioned each belong 
to two portions and so correspond to lines common to two sheets. It is by 
crossing these lines that a point passes from one sheet to another as the 
angle steadily increases. The point O in the w-plane is a winding point 
of the Riemann surface, its order is defined as the number n 1. 

A circle | w W \ = a n corresponds to a curve | z n Z n \ = a n , which 
belongs to the class of lemniscates* 

^2 ... r n = a n , 

where r x , r 2 , . . . r n are the distances of the point z from the points Z x , Z 2 , . . . Z n 
which correspond to W. In the present case the poles of the lemniscate 
are at the corners of a regular polygon and the equation of the lemniscate 
can be expressed in the form 

r 2n _ 2r n S n cos n (0 - 0) + R 2n = a 2n (re ie = z, Re i& = Z). 

When n = 2 a circle in the z-plane corresponds to a lima^on. To see 
this we write w = u + iv, z = x + iy, then 

u = x 2 y 2 y v = 2xy. 

* This is the name used by D. Hilbert, Gutt. Nachr. S. 63 (1897). The name cassinoid is used 
by C. J. de la Vallee Poussin, Mathesis (3), t. n, p. 289 (1902), Appendix. The geometrical pro- 
perties and types of curves of this kind are discussed by H. Hilton, Mess, of Math. vol. XLVIII, 
p. 184 (1919), reference being made to the earlier work of Serret, La Goupilliere and Darboux. 



270 Conformal Representation 

Hence, if . , 

(x + a}* + y 2 = c 2 , 

we have [u 2 4- v 2 - 2a 2 u + (a 2 - c 2 ) 2 ] 2 - 4c 4 (^ 2 + v 2 ), 

or, if /7 - u + c 2 - a 2 , F = v, U ^ RcosQ, V ^ R sin 0, 
([72 + 72 2c 2 Z7) 2 - 4a 2 c 2 (?7 2 + F 2 ), 
R = 2ac + 2c 2 cos0. 

EXAMPLES 

1. The curve r 2n 2r n c n cos n0 -f- c n d n has w ovals each of which is its own inverse 
with respect to a circle centre O and radius \/(cd). The ordinary foci B l9 B 2 ,...B n invert into 
the singular foci A 19 A 2 , ... A n , the polar co-ordinates of B s being given by r ^ d, nd ZSTT. 

2. Line charges of strength 4- 1 are placed at the corners of a regular polygon of n 
corners and centre 0, while line charges of strength 1 are placed at the corners of another 
regular polygon of n corners and centre O. Prove that the equipotentials are w-poled lemnis- 
cates. [Darboux and Hilton.] 

3. Prove also that the lines of force are n-poled lemniscates passing through the vertices 
of the regular polygons. 

4-22. The bilinear transformation. The transformation 

r ^ az b (A} 

^~ az+$> (A) 

in which a, 6, a, ]3 are complex constants, is of special interest because it 
is the only type of transformation which transforms the whole of the 
z-plane in a one-to-one manner into the whole of the -plane and gives a 
conformal mapping of the neighbourhood of each point. 

If a i- there are generally two points in the z-plane for which = z. 
These are given by the quadratic equation 

az 2 + (j8 - a) z - b - 0. 

Let us choose our origin in the z-plane so that it is midway between 
these points, then /? = a and if we write b = ac 2 the self-corresponding 
points are given by z = c. The transformation may now be written in 

theform +_c._a + ca + c 

c a caz ~ c ' 

From this relation a geometrical construction for the transformation 
is easily derived. Writing a + ca ^ ; 

+ c = RI&*I, z + c - r^'i, 
c = R 2 e l 2, z c = r 2 e ie z r 

R r 

we have the relations n 1 = p - 1 , 

^ 2 *> 

0! - 0, = CO + (^ - *,). 



The Bilinear Transformation 271 

If S l and S 2 are the self -corresponding points these relations tell us that 
a circle through S l and S 2 generally corresponds to a circle through S l 
and $ 2 , but in an, exceptional case it may correspond to a straight line, 
namely the line $!&,. 

Again, a circle which has S l and S 2 as inverse points corresponds to a 
circle which has $ x and S 2 as inverse points. 

By a suitable displacement of the z and -planes we can make any 
given pair of points the self-corresponding points provided the self-corre- 
sponding points &re distinct, for if the displacements are specified by the 
complex quantities u and v respectively, the transformation may be 
written in the form 



and we can choose u and v so that the equation = z has assigned roots 
z l and z 2 . 

We may conclude from this that the transformation maps any circle 
into either a straight line or a circle; a result which may be proved in 
many ways. One proof depends upon the theorem that in a bilinear 
transformation of type (A) the cross-ratio of four values of z is equal to 
the cross-ratio of the four values of ; i.e. 

(z- z,) (* 8 -_z 3 ) (r^j^Hk - k) 

(* - z 2 ) (* 3 - *i) ( - 2 ) (& - iV 

Now the cross-ratio is real when the four points lie on either a straight 
line or a circle, hence four points on a circle must map into four points 
which are either collinear or concyclic. If in the transformation (A) we 
choose u so that au + ft = 0, and v so that av = a, the transformation 

takes the form y 1 9 ,._ 

z=& 2 , ...... (B) 

where a 2 k 2 ab aft. 

By a suitable rotation of the axes of reference we can reduce the 
transformation to the case in which k is real, and this is the case which 
will now be discussed. 

The transformation evidently consists of an inversion in a circle of 
radius k with centre at the origin followed by a reflection in the axis of x. 
The points z = k are self-corresponding points and if these are denoted 
by S 1 and S 2 it is easily seen that two corresponding points P and Q lie 
on a circle through S^ and S 2 . The figure has a number of interesting 
properties which will be enumerated. 

1. Since z ( -f k) = k (z -f k) the angles /S X PO, QS^ are equal, and 
so the angles S 1 PO, S 2 PQ are also equal. 

2. The triangles S 1 PO, QPS% are similar, and so 

PSi.PSt~PO.PQ. 



272 Conformal Representation 

3. If C is the middle point of PQ we have CS^CS^ = CP 2 , also PQ 
bisects the angle S^CS 2 . 

The four points S l9 P, S 2 , Q on the circle form a harmonic set. This 
follows from the relation 

1 4- _ =-r 

z_ k^z + k z- V 

which is easily derived from (B). 

The angle PS^C is equal to the angle 8^0. The lines ^P, P0, (7^ 
thus form an isosceles triangle. 

In the case when the self-corresponding points coincide we have 

a - fl = 2ac, b = - ac 2 , 

where c is the self-corresponding point. The transformation may now be 
written in the form 

~ = Q -"- --+ , + ac * 0. 
c ft + ac z ~ c r 

It may be built up from displacements and transformations of the 
type just considered and so needs no further discussion. The only other 
interesting special case is that in which the transformation then consists 
of a displacement followed by a magnification and rotation. 

4-23. Poissorts formula and the mean value theorem. Bocher has 
shown by inversion that Poisson's formula may be derived from Gauss's 
theorem relating to the mean value of a potential function round a circle. 

Let C and C' be inverse points with respect to the circle F of radius a, 
and let CO' = c. Inverting with respect to a circle whose centre is C' and 
radius c, the point 8 on the circle transforms into a point 8'. We shall 
suppose that C' is outside the circle F, then S is inside the circle F. Let 
ds y els' be corresponding arcs at 8 and 8' respectively and let the polar 
co-ordinates of C and C 1 be (r, 6), (/, 0) respectively, where rr' = a 2 . The 
circle F inverts into a circle with centre C and radius given by the formula 
aa' - or, for rR . 

PS' ~ r - r 

C - C C 'S CM 

a r cr , 

= c -, - = = a . 
r a a 

Also r*C'S* - a*.CS 2 - a 2 [r 2 -fa 2 - 2ar cos (a - 0)], 

where (a, a) are the polar co-ordinates of 8. 
Writing ds' ~ a'dv , we have 

, , _ c 2 ds ca 2 da __ rcda 

= a'T C"& ^ r ". C'S* = a 2 - 2ar cos~(a - "^"Tr 5 ' 
and re = a 2 r 2 , consequently the formula of Poisson becomes 



Circle and Half Plane 273 

This formula states that the mean value of a potential function round 
the circumference of a circle is equal to the value of the function at the 
centre of the circle. Hence Poisson's formula may be derived from this 
mean value theorem and is true under the same conditions as the mean 
value theorem. 

4-24. The conformal representation of a circle on a Jialf plane*. If two 
plane areas A and A can be mapped on a third area A Q they can be 
mapped on one another, eonseqiiently the problem of mapping A on A t 
reduces to that of mapping A and A l on some standard area A . 

This standard area A Q is generally taken to be either a circle of unit 
radius or a half plane. The transition from the circle x 2 -f y 2 < 1 in the 
z-plane to the half plane v > in the ?#-plane is made by means of the 
substitutions 

z = x -f iy, w = u -j- iv, z (i -j- w) = i w, 

Dx - 1 - u 2 - v 2 , l)y - 2i/, (I) 

where D = u 2 {- (1 -f- v) 2 , 

4/D - (1 4- x) 2 -f y 2 . 

When v = 0, the substitution u = tan 6 gives 
x = cos 20, y = sin 29. 

As 26 varies from TT to TT, the variable u varies from oo to oo and 
so the real axis in the w-plane is mapped in a uniform manner on the unit 
circle x 2 -f- y 2 = 1 in the z-plane. 

Since, moreover, 

D (1 - x 2 - y 2 ) = 4?;, Dy = 2u 9 

we have v > when x 2 -f y 2 < 1, consequently the interior of the circle is 
mapped on the upper half of the w-plane. 

When u and v are both infinite or when either of them is infinite, we 
have x = 1, y = 0; hence the point at infinity in the w-plane corre- 
sponds to a single point in the z-plane and this point is on the unit circle. 

The transformation (I) may be applied to the whole of the z-plane; 
it maps the region outside the circle x 2 + y 2 = 1 on the lower half (v < 0) 
of the w-plane. A line y = mx drawn through the centre of the circle 
corresponds to a circle m (1 u 2 v 2 ) = 2u whjch passes through the 
point (0, 1) which corresponds to the centre of the circle, and through the 
point (0, 1) which corresponds to the point at infinity in the z-plane. 
This circle cuts the line v = orthogonally. 

Two points which are inverse points with respect to the circle x 2 -j- y 2 = 1 
map into points which are images of each other in the line v = 0. 

* This presentation in 4-24, 4-61 and 4-62 follows closely that given in Forsyth's Theory of 
Functions and the one given in Darboux's Thdorie generate des surfaces, 1. 1, pp. 170-180. 
B 18 



274 Conformal Representation 

The upper half of the w-plane may be mapped on itself in an infinite 
number of ways. To see this, let us consider the transformation 

Y aw + b dt, b /T1A 

= -7-J9 W = =, (II) 

cw -f d a c 

in which a, 6, c and d are real constants and = -f irj. 

When w is real is also real and vice versa, hence the real axes corre- 
spond. Furthermore, 

7? [(cu + d) 2 + c 2 v 2 ] = (ad- be) v, 

hence Had be is positive, T? is positive when v is positive. There are three 
effective constants in this transformation, namely, the ratios of a, b and c 
to d, hence by a suitable choice of these constants any three points on the 
axis of u may be mapped into any three points on the axis of . In fact, 
if u ly u 2 , u s are the values of u corresponding to the values g l9 2 , 3 f l> 
we can say from the invariance of the cross-ratio that 

( - li) (& ~ la) _ ( w ~ ^i) fa? - ^a) 
( - sY) (& ~ ~ li) (^ -^2) fas ~ ^i) ' 

and so the equation of the transformation may be written down in the 

previous form, the coefficients being 

a = && fa 2 ~ ^3) + fall faa - %) 4- Iil 2 fai ~ ^2), 

& - ^ 3 ) + Wa^ila (la - li) + ^1^2 Is (li I 2 ) 
| 3 ) + ^2 (la ~ li) + ^ 3 (li ~ l a ) 

- ^ 3 ) + 7^3^ (| 3 - ^) + U 1 U 2 (^ - f 2 ). 

The quantity ad 6c is given by the formula 
ad - be - ( 2 - &) (& - ^) (^ - f 2 ) (^ 2 - u 3 ) (u^ - %) (^ - u 2 ). 

If i^, u 2 , t6 3 are all different the coefficients c and d cannot vanish 
simultaneously, for the equations c = 0, d = give 

b2 S3 S3 "" b 1 _ b 1 ~~ b 2 

1 ( V - % 2 ) "" ^2 (^3 2 ~ % 2 ) ~ ^3 (^l 2 - ^2 2 ) ' 

and these equations imply that 

*i ( W 2 2 - % 2 ) + ^2 (^3 2 ~ % 2 ) + ^3 (% 2 - ^ 2 2 ) = 0, 
or (w 2 - wj) (i^ - Ul ) (HI - ^ 2 ) = 0, 

if the quantities f x , ^ 2 , ^ 3 are also all different. In a similar way it can be 
shown that a and c cannot vanish simultaneously and that a and b cannot 
vanish simultaneously. Poincar6 has remarked that the transformation 
(II) can usually be determined uniquely so as to satisfy the requirement 
that an assigned point and an assigned direction through this point 
should correspond to an assigned point w and an assigned direction 
through this point. The proof of the theorem may be left to the reader, 



Riemanrfs Problem 275 

who should examine also the special case in which one or both of the 
points is on the real axis in the plane in which it lies. 

EXAMPLES 

1. Prove that Poisson's formula 

7 = 2* j _. T-2rcos(6~-'iJ')~-rr* \ r \ <l 

maps into the formula of 3*11, 

_ 1 f = yF (x'} dx' 

~ TT / -co (x -x') 2 + y*' 
where / (0') = F (tan 0'). 

2. Prove that the transformation 



maps the half plane y > on the unit circle | w | < 1 in such a way that the point z maps 
into the centre of the circle. 

4-31. Riemann's problem. The standard problem of conformal repre- 
sentation will be taken to be that of mapping the area A in the z-plane 
.on the upper half of the w-plane in such a way that three selected points 
on the boundary of A map into three selected points on the axis of u. This 
is the problem considered by B. Riemann in his dissertation. The problem 
may be made more precise by specifying that the function / (w) which 
gives the desired relation 

z=f(w) 

should possess the following properties : 

(1) / (w) should be uniform and continuous for all values of w for which 
v > 0. If W Q is any one of these values/ (w) should be capable of expansion 
in a Taylor series of ascending powers of w W Q which has a radius of 
convergence different from zero. 

(2) The derivative/' (w) should exist and not vanish for v > ; indeed, 
if /' (w) = for w = W Q there will be at least two points in the neighbour- 
hood of w for which z has the same value. This is contrary to the require- 
ment that the representation should be biuniform. 

(3) / (?#) should be continuous also for all real values of tv, but it is 
not required that in the neighbourhood of one of these values, W Q , the 
function / (iv) can be expanded in a Taylor series of ascending powers of 
w W Q) for, as far as the mapping is concerned, / (w) is defined only 
for v > 0. 

(4) Considered as a function of z, the variable w should satisfy the 
same conditions as/ (w). If/ (w) satisfies all these requirements it will give 
the solution of the problem. The solution is, moreover, unique because if 

two functions , , . , . 

z = f(w), z = g(w) 

give different solutions of the problem, the transformation 

18-2 



276 Conformal Representation 

will map the upper half plane into itself in such a way that the points 
^i > u z ) U 3 m &p into themselves. Now it can be proved that a transformation 
which maps the upper half plane into itself is bilinear and so the relation 
between w and W is 

(w - uj (u 2 - u 3 ) ^ (W -7^) (w_ 2 _- ?/ 3 ) 

(w - u 2 ) (u 3 - uj ' (W - u 2 ) (w. a - u^ 

w HI W u, 

mr A A 

ui . . . 

W U 2 W U 2 

This reduces to 

(11, - W) (u, - u 2 ) - 0, 
and so W = w. 

4-32. TAe (jeneral problem of conformed representation. The general type 
of region which is considered in the theory of conformal representation 
may be regarded as a carpet which is laid down on the z -plane. This carpet 
is supposed to have a boundary the exact nature of which requires careful 
specification because with the aim of obtaining the greatest possible 
generality, different writers use different definitions of the boundary curve. 
There may, indeed, be more than one boundary curve, for a carpet may, 
for instance, have a hole in its centre. For simplicity we shall suppose 
that each boundary curve is a simple closed curve composed of a finite 
number of pieces, each piece having a definite direction at each of its 
points. At a point where two pieces meet, however, the directions of the 
two tangents need not be the same; a carpet may, for instance, have a 
corner. The tangent may actually turn through an angle 2?r as we pass 
from one piece of a boundary curve to another and in this case the boundary 
has a sharp point which may point either inwards or outwards. It turns 
out that the fojmer case presents a greater difficulty than the latter. 

In special investigations other restrictions may be laid on each piece 
of a boundary curve and from the numerous restrictions which have beer 
used we shall select the following for special mention. 

(1) The direction of the tangent is required to vary continuously as 
a point moves along the curve (smooth curve*). 

(2) The curvature is required to vary continuously as a point moves 
along the curve. 

(3) The curve should be rectifiable, i.e. it should be possible to define 
the length of any portion of the curve with the aid of a definite integral 
which has a precise meaning specified beforehand, such as the meaning 
given to an integral by Riemann, Stieltjes or Lebesgue. 

A simple curve which possesses the first property may be called a 
curve (CT), one which possesses the properties 1 and 2 a curve (CTC), 
a curve which possesses the properties 1 and 3 may be called a curve (RCT). 

* A curve which is made up of pieces of smooth curves joined together may be called smooth 
bit by bit ("Stiickweise glatte Kurve"; see Hurwitz-Courant, Berlin (1925), Funktionentheorie). 



Properties of Regions 277 

The carpet will be said to be simply connected when a cross cut starting 
from any point of the boundary and ending at any other divides the carpet 
into two pieces. A carpet shaped like a ring is not simply connected 
because a cut starting from a point on one boundary and ending at a point 
on the other does not divide the carpet into two pieces. ^Such a carpet 
may, however, be made simply connected by making a cut of this type. 
When we consider a carpet with n boundaries which are simple closed 
curves we shall suppose that the boundaries can be converted into one 
by a suitable number of cuts which will at the same time render the carpet 
simply connected. It will be supposed, in fact, that the carpet is not 
twisted like a Mobius' strip when the cuts have been made. 

Any closed curve on a simply connected carpet can be continuously 
deformed until it becomes an infinitely small circle. This cannot always be 
done on a ring-shaped carpet as may be seen by considering a circle con- 
centric with the boundary circles of a ring, and it cannot be done in the 
case of a curve which runs parallel to the edge of a singly twisted Mobius' 
strip formed by joining the ends of a thin rectangular strip of paper after 
the strip has been given a single twist through 180. Such a closed curve 
is said to be irreducible and the connectivity of a carpet may be defined 
with the aid of the number of different types of irreducible closed curves 
that can be drawn on it. Two closed curves are said to be of different types 
when one cannot be deformed into the other without any break or crossing 
of the boundary. It is not allowed, for instance, to cut the curve into 
pieces and join these together later or in any way to make the curve into 
one which does not close. 

A simply connected carpet may cover the plane more than once; it 
may, for instance, be folded over, or it may be double, triple, etc. In the 
latter case it is called a Riemann surface, i.e. a surface consisting of several 
sheets which connect with one another at certain branch lines in such a 
way as to give a simply connected surface. When there are only two 
sheets it is often convenient to regard them as the upper and lower 
surfaces of a single carpet with a cut or branch line through which passage 
may be made from one surface to the other. In the case of a ring-shaped 
carpet we generally consider only the upper surface, but if the lower surface 
is also considered and a passage is allowed from one surface to the other 
across either one or both of the edges of the ring a surface with two sheets 
is obtained, but this doubly sheeted surface is not simply connected 
because a curve concentric with the two edges is still irreducible. In a more 
general theory of conform al representation the mapping of multiply 
connected siirf aces is considered, but these will be excluded from the present 
considerations . 

A carpet may also have an infinite number of boundaries or an infinite 
number of sheets, but these cases will also be excluded. When we speak of 



278 Conformal Representation 

an area A we shall mean the right side of a carpet which is bounded by a 
simple closed curve formed of pieces of type (RCTC) and is not folded 
over in any way. The carpet will be supposed, in fact, to be simply con- 
nected and smooth, the word smooth being used here as equivalent to the 
German word "schlicht," which means that the carpet is not folded or 
wrinkled in any way. The function F (z) maps the circular area | z \ < 1 
into a smooth region if 

F fa) - F (z 2 ) 

- 1 * ' 

whenever | z l \ < 1 and | z 2 | < 1 . 

We shall be occupied in general with the conformal representation of 
one simple area on another, and for brevity we shall speak of this as a 
mapping. In advanced works on the theory of functions the problem of 
conformal representation is considered also for the case of Riemann 
surfaces and the more general theory of the conformal representation of 
multiply connected surfaces is treated in books on the differential geometry 
of surfaces. 

For many purposes it will be sufficient to consider the problem of 
conformal representation for the case of boundaries made up of pieces of 
curves having the property that the co-ordinates of their points can be 
expressed parametrically in the form 



where the functions / (t) and g (t) can be expanded in power series of type 

2a n (*-* ) n , ...... (HI) 

n-O 

which are absolutely and uniformly convergent for all values of the 
parameter t that are needed for the specification of points on the arc under 
consideration. In such a case the boundary is said to be composed of 
analytic curves and this is the type of boundary that was considered in 
the pioneer work of H. A. Schwarz, but the restriction of the theory to 
boundaries composed of analytic curves is not necessary* and a method 
of removing this restriction was found by W. F. Osgoodf. His work has 
been followed up by that of many other investigators^. 

In the power series (III) the quantities t Q , a n are constants which, of 
course, may be different for different pieces of the boundary. 

There are really two problems of conformal representation. In one 
problem the aim is simply to map the open region A bounded by a curve 

* In the modern work the boundary considered is a Jordan curve, that is, a curve whose points 
may be placed in a continuous (1,1) correspondence with the points of a circle. 
t Trans. Amer. Math. Soc. vol. i, p. 310 (1900). 
t Particularly E. Study, C. Caratheodory, P. Koebe and L. Bieberbach. 



Exceptional Cases 279 

a on the open region B bounded by a curve 6, there being no specified 
requirement about the correspondence of points on the two boundaries. 
In the second problem the aim is to map the closed realm* A on the 
closed realm B in such a way that each point P on a corresponds to only 
one point Q which is on b and so that each point Q on b corresponds to 
only one point P which is on a. It is this second problem which is of most 
interest in applied mathematics. If, moreover, in the first problem the 
correspondence between the boundaries is not one-to-one the applied 
mathematician is anxious to know where the uniformity of the corre- 
spondence breaks down. 

Existence theorems are more easily established for the first problem 
than for the second and fortunately it always happens in practice that 
a solution of the first problem is also a solution of the second ; but this, of 
course, requires proof and such a proof must be added to an existence 
theorem that is adapted only for the first problem. 

The methods of conf ormal representation are particularly useful because 
they frequently enable us to deduce the solution of a boundary problem 
for one closed region A from the solution of a corresponding boundary 
problem for another region B which is of a simpler type. When the function 
which effects the mapping is given by an explicit relation the process of 
solution is generally one of simple substitution of expressions in a formula, 
but when the relation is of an implicit nature or is expressed by an infinite 
series or a definite integral the direct method of substitution becomes 
difficult and a method of approximation may be preferable. A method of 
approximation which is admirable for the purpose of establishing the 
existence of a solution may not be the best for purposes of computation. 

4-33. Special and exceptional cases. It is easy to see that it is not 
possible to map the whole of the complex z-plane on the interior of a circle. 
Indeed, if there were a mapping function / (z) which gave the desired 
representation, / (z) would be analytic over the whole plane and | / (z) \ 
would always lie below a certain positive value determined by the radius of 
the circle into which the z-plane maps, consequently by Liouville's theorem 
/ (z) would be a constant. A similar argument may be used for the case 
of the pierced z-plane with the point z as boundary. By means of a 
transformation z z = 1/z' the region outside z can be mapped into the 
whole of the z'-plane when the point z' = oo is excluded. The mapping 
function is again an integral function for which \f(z')\<M, and is thus 
a constant. On account of this result a region considered in the mapping 
problem is supposed to have more than one external point, a point on the 
boundary being regarded as an external point. 

* We use realm as equivalent to the German word " Bereich, " and region as equivalent to 
" Gebiet." 



280 Conformal Representation 

The next case in order of simplicity is the simply connected region 
with at least two boundary points A and B. If these were isolated the 
region would not be simply connected. We shall therefore assume that 
there is a curve of boundary points joining A and B. This curve may 
contain all the boundary points (Case 1) or it may be part of a curve of 
boundary points which may either be closed or terminated by two other 
end-points C and F. The latter case is similar to the first, while the case 
of a closed curve is the one which we wish eventually to consider. 

The simplest example of the first case is that in which the end-points 
are z -= and z = oo, the boundary consisting of the positive x-axis. The 
region bounded by this line can be regarded as one sheet of a two-sheeted 
Riemann surface with the points and oo as winding points of the second 
order, passage from one sheet to the other being made possible by a 
junction of the sheets along the positive #-axis. The whole of this Riemann 
surface is mapped on the z' -plane by means of the simple transformation 
z' y'z, which sends the one sheet in which we are interested into the 
half plane < 6' < 77, where z' = r'e lQf . 

In the case when the boundary consists of a curve joining the points 
z = a, z =- 6, these points are regarded as winding points of the second 
order for a two-sheeted Riemann surface whose sheets connect with each 
other along the boundary curve. This surface is mapped on the whole 
z'-plane by means of the transformation 



' ( Z ~ "}* cz' ~ 1 
( z -~b) - cz - *> 



and in this transformation one sheet goes -into the interior, the other into 
the exterior of a certain closed curve (7. The mapping problem is thus 
reduced to the mapping of the interior of C on a half plane or a unit circle. 
Finally, by means of a transformation of type 

z= Az' + B, 

we can transform the region enclosed by C into a region which lies entirely 
within the unit circle | z \ < 1 and our problem is to map this region on 
the interior of the unit circle | | < 1 by means of a transformation of 
type J=/(z). 

4-41. The mapping of the unit circle on itself. If a and a are any two 
conjugate complex quantities and a is a real angle the quantities e* a a 
and 1 de la have the same modulus, consequently if )3 is another real 
angle the transformation 

(1 -&) = e*(z-a) (A) 

maps | z | = 1 into | | = 1, and it is readily seen that the interior of one 
circle maps into the interior of the other. It should be noticed that this 



Circle Mapped on Itself 281 

transformation maps the point z = a into the centre of the circle | | = 1, 
If we put a = the transformation reduces to the rotation 

= ze*, 

which leaves the centre of the circle unaltered. If we can prove that this 
is the most general conformal transformation which maps the interior of 
the unit circle into itself in such a way that the centre maps into the 
centre it will follow that the formula (A) gives the most general trans- 
formation which maps the unit circle into itself. 

The following proof is due to H. A. Schwarz. 

Let / (z) be an analytic function of z which is regular in the circle 
| z | = 1 and satisfies the conditions 

\f(z)\< 1 for \z\< 1, /(0) = 0. 

If "<(*) =/(*)/*> </>(<>) =/'(0), 

the function </> (z) is also regular in the unit circle, and if | z \ = r, where 
r < I, 1 we have 

I < (z) I < l/r- 

But since </> (z) is analytic in the circle | z \ = r the maximum value 
of | <f> (z) | occurs on the boundary of this region and not within it, hence 
for a point z within the circle | z \ = r, or on its circumference, we have 
the inequality | <f> (z ) | < l/r (Schwarz's inequality*). 

Passing to the limit r -> 1 we have the inequality 
| < (z ) | < 1 for | z | < 1. 

Now let = / (z), z -= g () be the mapping functions which map a 
circle on itself in such a way that the centre maps into the centre, then 
by Schwarz's inequality 

| / z | < 1, | z / | < 1. 

Rence | /2 | == 1, and so | </> (z) \ is equal to unity within the unit 
circle. Now an analytic function whose modulus is constant within the 
unit circle is necessarily a constant, hence = ze l ? where j8 is a constant 
real angle. 

Since the unit circle is mapped on a half plane by a bilinear trans- 
formation, it follows that a transformation which maps a half plane into 
itself is necessarily a bilinear transformation. 

4-42. Normalisation of the mapping problem. Let F be the unit circle 
| | < 1 in the -plane and suppose that a smooth region G in the 
z-plane can be mapped in a (1, 1) manner on the interior of F. Since F 
can be mapped on itself by a bilinear transformation in such a way that 
two prescribed linear elements correspond, it is always possible to normalise 

* This is often called Schwarz's lemma as another inequality is known as Schwarz's inequality. 
The lemma of 4*61 is then called Schwarz's principle or continuation theorem. This second in- 
equality is used in 4-81. 



282 Conformal Representation 

the mapping so that a prescribed linear element in the region corresponds 
to the centre of the unit circle and the direction of the positive real axis, 
that is to what we may call the " chief linear element." We can then, 
without loss of generality, imagine the axes in the z-plane to be chosen so 
that the origin lies in on the prescribed linear element and so that this 
linear element is the chief linear element for the z-plane. This means that 
the normalised mapping function = / (z) satisfies the conditions/ (0) = 0, 
/' (0) > 0. Finally, by a suitable choice of the unit of length in the z-plane, 
or by a transformation of type z' = kz, we can make/' (0) = 1. The trans- 
formation is then fully normalised and / (z) is a completely normalised 
mapping function. The power series which represents the function in the 
neighbourhood of z = is of type 

/(z) - z-f a 2 z 2 + .... 

The coefficients a 2 > a s> * n this series are not entirely arbitrary, in 
fact it appears that a 2 is subject to the inequality* | a 2 \ < 2. To prove this 
we consider the function g (z) defined by the equation / (z) g (z) = 1 . We 



If 0<c< 1, the transformation y = g (z) maps the circular ring 
c < z < 1 on a region A in the y-plane bounded by a curve C and a curve 
C c which can be represented parametrically by the equation 

cy = e-* -f 6 x c + & 2 cV* + c 3 o> (c, a), 

where a is the parameter and co (c, a) remains bounded as a varies between 
and 277. Writing 6 2 = be 2 *, cd = 1, we remark that the equation 



^1 V 

cJO 



gives the parametric representation of an ellipse with semi-axes d + be, 
d be respectively, and so the area A c of the curve C e differs from nd 2 
by cB, where | B \ remains bounded as c -* 0. Now the area of the region A 
is a quantity A c ' given by the equation 

-l+ S n | b n I 2 - S n \ b n I 2 c 

n-2 n-2 

and A c > A c '-, also as c -*> the difference nd* A c ' tends to the area A 
of the region enclosed by C. Since A > we have the inequality 

S n\b n \*<\. 

n-2 

Now the function 

[<7 (z 2 )]* = ^ -f ftz+..., 20!=^ 

likewise maps the unit circle | z | < 1 on a smooth region, and so by the 

last theorem , Q , 

\ Pi\ < ! 
* See, for instance, L. Bieberbach, Berlin. Sitzungsber. Bd. xxxvm, S. 940 (1916). 



Inequalities 283 

Since b l = a 2 the last inequality becomes simply 
| 2 | 2 < 4 or | a 2 | < 2. 

We have | & | = 1 when and only when /? 2 = /? 3 = ... = 0, consequently 
| a 2 | = 2 when and only when 

[g (z 2 )]^ = - -f ze icr , where a is real, 

z 

or gr(z) = (z~* + eM) 2 , 

that is, when / (z) = ( L -*^ . 

EXAMPLES 

1. A transformation which maps the unit circle into itself in a one-to-one manner and 
transforms the chief linear element into itself is necessarily the identical transformation. 

[Schwarz, and Poincare".] 

2. A region enclosing the origin which can be mapped on itself with conservation of the 
chief linear element consists either of the whole plane or of the whole plane pierced at the 
origin. [T. Rad6, Szeged Acta, 1. 1, p. 240 (1923).] 

3. If the region | z \ < 1 is mapped smoothly on a region W in the w- plane by the function 
w =f(z) = z + ajz 2 + agZ 3 -f- ..., prove that, when | z \ = r < 1, 



Hence show that if W Q is a point not belonging to the region W 

I w o I < i- 
The value | w \ = J is attained at the point w = Je"*" when 



4. If a region W of the w-plane is mapped smoothly on the circle | z \ < 1 by the f unction 
w =/(z) and if Z 19 Z 2 are any two points which do not lie within the circle, then 



[G. Pick, Leipziger Berichte, Bd. LXXXJ, S. 3 (1929).] 

4-43. The derivative of a normalised mapping function. Now let / (z) 
be regular in the unit circle | z | < 1, which we shall call K. We shall study 
the behaviour of /' (z) in the neighbourhood of an interior point z of K. 
Let z be the conjugate of z , then the transformation 

z' - Z ~ * 

1 ZZ 

maps the circle K into itself and sends the point z into the point z' = 0. 
Thus 



284 



Conformal Representation 

/*( z )=/(i z +-! 2 )-/( z <>)> 



Writing 

we see that the function /* (z) maps the circle K on a smooth region and 
leaves the origin fixed. If z = re ie we have by Taylor's theorem 

/* (z) - z (1 - r)/' (z ) + 2 (1 - ^) [(1 - r 2 ) /" fo) - 2z /' (z )] + - 
The function! 

*M-<.^ta>-' + ^ 1+ "- 

is thus a normalised mapping function, and so by the theorem of 4-42, 
I A 2 | < 2, i.e. 

'(1 



Writing z J v ; o; - P + iQ - r ^ (w -1- iv), 

where/' (2) e u + tv and ^ and v are real, we have the inequality 

4r 



Therefore 



4 - 2r 



: ]_ , 

4 -f 2r 

4 



4 3v 
"" 1 - r a< 3r ^ 1 - r 2 " 

Integrating between and r we obtain the twa inequalities J 

1 - r ^ ^ , I -f r 



log 



( 1 ~ 



r Fhe first of these may be written in the more general form 



. 

(1+ |z|)'~ 



/' (z) 
/'(O) 



1 I- 
(1 - 



where now ^ = f (z) is a function which maps K on a smooth region not 

f This function was used by L. Bicberbach, Math. Zeitschr. Bd. iv, S. 295 (1919), and later by 
R. Nevanlinna, see Bieberbach, Math. Zeitechr. Bd. ix, 8. 161 (1921). The following analysis 
which is due to Nevanlinna is derived from the account in Hurwitz-Courant, Funktionentheone, 
S. 388 (1925). 

t The first of these was given by T. H. Gronwall, Comptes Rendus, t. CLXII, p. 249 (1916), 
and by J. Plemelj and G. Pick, Leipziger Ber. Bd. LXVIII, S. 58 (1916). In the form 

k(r)<\f'(z)\ <h(r) 

it is known as Koebe's Verzerrungssatz (distortion theorem). The precise forms for k (r) and h (r) 
were derived also by G. Faber, Munchener Ber. S. 39 (1916) and L. Bieberbach, Berliner Ber. 
Bd. xxxvm, S. 940 (1916). The inequality satisfied by | v \ was discovered by Bieberbach, Math. 
Zeitschr. Bd. iv, S. 295 (1919). 



Properties of the Derivative 285 

containing the point at infinity but is not necessarily a normalised mapping 
function. 

The second theorem is called the rotation theorem, as it indicates limits 
for the angle through which a small area is rotated in the conformal 
mapping. The other theorem gives limits for the ratio in which the area 
changes in size. This theorem has been much used by Koebe* in his 
investigations relating to the conformal representation of regions and has 
been used also in hydrodynamics^ and aerodynamics. 

When / (z) maps K on a convex region it can be shown that | /' (z) \ 
lies within narrower limitsj. Study has shown, moreover, that in this case 
any circle within K and concentric with it also maps into a convex region. 
Many other inequalities relating to conformal mapping are given in a paper 
by J. E. Littlewood, Proc. London Math. Soc. (2), vol. xxm, p. 481 (1925). 

4-44. The mapping of a doubly carpeted circle with one interior branch 
point. Let P be a point within the unit circle | z' \ < 1 and let r 2 e ld 
(0 < r < 1) be the value of z' at P. The transformation || 

(1 + r 2 )z- 2re* . , x ,.. 

z = *2 (i^r'je*'^ (A) 

satisfies the conditions </> (0) = 0, </>' (0) > 0, | z' \ = \ z j, when z \ =- 1, and 
so represents a partially normalised transformation which maps the unit- 
circle in the z-plane on a doubly carpeted unit circle in the z'-plane, the 
two sheets having a junction along a line extending from P to the boundary. 
We shall regard this line as a cut in that sheet which contains the chief 
element corresponding to the chief element in the z-plane. 

It is evident that | z' \ < \ z \ whenever | z \ < 1 , and so | z \ < 1 when 
| z | < 1. This means that | z \ < \z\ whenever | z' \ < 1. 

From this we conclude that for all values of z' for which | z' \ < r 2 there 
is a positive number q (r) greater than unity for which | z \ > q (r) \ z' \ . 
Indeed, if there were no such quantity g (r) there would be at least one 
point in the circle | z' \ < r 2 , for which z \ = | z' \ . An expression for q (r) 
may be obtained by writing 

2r 

'-l + r" * = ?*. 

and considering the points s, l/s on the real axis. If E l , R 2 are the distances 
of the point z from these points respectively we have 

I 'I- -at 

* Gott. Nachr. (1909); Crdle, Bd. cxxxvm, S. 248 (1910); Math. Ann. Bd. LXIX. 

f Ph. Frank and K. Lowner, Math. Zeitschr. Bd. in, S. 78 (1919). 

t T. H. Gronwall, Comptes Rendus, t. CLXII, p. 316 (1916). 

E. Study, Konforme Abbildung einfazh zusammenhiingender Bereiche, p. 110 (Teubner, 
Leipzig, 1913). A simple proof depending on a use of Schwarz's inequality has been given recently 
by T. Rado, Math. Ann. Bd. en, S. 428 (1929). 

H C. Caratheodory, Math. Ann. Bd. Lxxn, S. 107 (1912). 



286 Conformal Representation 

The oval curve for which pR 1 /sR 2 = r 2 lies entirely within a circle 
R 2 /R 1 = constant which touches it at a point # = p on the real axis 
for which / , \ 2 



ps), 



r 



l + r , [(2 + 2r*)i - 1 + r]. 



The constant is found to be 



and we may take this as our value of q (r). We can see that it is greater 
than 1 when r < 1 because 

2 f 2r 4 - (1 -f r - r 2 + r 3 ) 2 - (1 - r 4 ) (1 - r) 2 . 

It is clear from the inequality | z \ > q (r) \ z' \ that points corre- 
sponding to those which lie within the circle | z \ < r 2 in the z' -plane lie 
within the larger circle | z | < r 2 q (r) in the z-plane. If r 2 is the minimum 
distance from the origin of points on a closed continuous curve C" which 
lies entirely within the unit circle | *' | < 1, the transformation (A) maps 
the interior of C' into the interior of a closed* curve C which lies entirely 
between the two circles | z \ < 1, | z \ = r 2 q (r). The shortest distance "from 
the origin to a point of C may be greater than r 2 q (r) but it lies between 
this quantity and r, i.e. the value of | z | corresponding to the branch 
point z = e ie r 2 . This second minimum distance may be used as the constant 
of type r 2 in a second transformation of type (A). Let us call it r x 2 and 
use the symbol C l to denote the curve into which C is mapped by the new 
transformation. The minimum distance from the origin of a point of this 
curve is a quantity r 2 2 which is not less than a quantity r-^q (r x ) associated 
with the number r x . 

If we consider the worst possible case in which the minimum distance 
for a curve C n+l derived from a curve C n with minimum distance r n 2 is 
always r n z q (r n ), we have a sequence of numbers r l9 r 2 , ... r n , which -are 

derived successively by means of the recurrence relation 



W = ,-^V, [(2 + 2r.)* - 1 + r n ]. 

i ~r r n 

Since r n < 1 for all values of n and r n4l > r n , the sequence tends to 
a limit R which must be given by the equation 

R* = -Jj-fr [(2 + 25*)* - 1 + R*l 

This equation gives the value R = 1. Hence as n ->oo the curve C n lies 
between two circles which ultimately coincide. 

* The curve C may in some cases close by crossing the line which corresponds to the cut in 
the z'-plane. This will not happen if the cut is drawn so that it does not intersect C' again. 



Sequence of Mapping Operations 287 

The convergence to the limit is very slow, as may be seen by considering 
a few successive values of r n 2 : 

r 2 r 2 r 2 

r l r 2 r 3 

25 -283 -309 

The best possible case from the point of view of convergence is that in 
which /i 2 = r. This case occurs when the curve C" is shaped something like 
a cardioid with a cusp at P. 

Though useful for establishing the existence of the conformal mapping 
of a region on the unit circle, the present transformation is not as useful 
as some others for the purpose of transforming a given curve into another 
curve which is nearly circular, unless the given curve happens to be shaped 
something like a cardioid or a Iima9on with imaginary tangents at the 
double point. We shall not complete the proof of the existence of a 
mapping function for a region bounded by a Jordan curve. This is done 
in books on the theory of functions such as those of Bieberbach and 
Hurwitz-Courant. Reference may be made also to the tract on conformal 
transformation which is being written by Caratheodory*, to E. Goursat's 
Cours d' Analyse Mathematique, t. in, and to Picard's Traite d' Analyse. 

4-45. The selection theorem. Let us suppose that the set or sequence 
of functions u (x, y), u 2 (x, y), u 3 (x, y), ... possesses the following pro- 
perties : 

(1) It is uniformly bounded. This means that in the region of definition 
E the functions all satisfy an inequality of type 

I u 8 (x, y) | < M , 

where M is a number independent of s and of the position of the point 
(x, y) of the region R. 

(2) It is equicontinuous^ . This means that for any small positive number 
c there is an associated number S independent of s, x and y but depending 
on in such a way that whenever 

(x' - xY + (y r - y) 2 < S 2 
we have | u 8 (x' ', y') - u s (x, y) \ < \c. 

We now suppose that the sequence contains an unlimited number of func- 
tions and that an infinite number of these functions forming a subsequence 
^wi ( x > y)> U m2 ( x > y}> can be selected by a selection rule (m). Our aim 
now is to find a sequence (1), (2), (3), ... of selection rules such that the 
"diagonal sequence " u n (x, y), u 22 (x, y), ... converges uniformly in R. 

* Carathe"odory's proof is given in a paper in Schwarz- Festschrift, and in Math. Ann. Bd. LXXII, 
S. 107 (1912)., Koebe's proof will be found in his papers in Journ. ftir Math. Bd. CXLV, S. 177 
(1915); Acta Math. t. XL, p. 251 (1916). 

f The idea of equal continuity was introduced by Aecoli, Mem. d. R. Ace. dei Lincei, t. xvm 
(1883). 



288 Conformed Representation 

The first step is to construct a sequence of points P x , P 2 , ... everywhere 
dense in R. This may be done by choosing our origin outside R and using 
for the co-ordinates of P s expressions of type 

X B = p2~", y,= p'2-" 9 q>0 

where p, p' and q are integers, and where the index s = / (p, p', q) is a 
positive integer with the following properties: 

1 (P> P r ><l)> I (Po> Po'> 9o)> whenever q > q , 
I (P> />' ?) > I (Po> Po'> 7), whenever p > p , 
L (P, P', ( l) > l (P> Po, ?) whenever p' > p '. 

Since the functions M S (a;, y) are bounded, their values at P l have at least 
one limit point U l (x l , y^. We therefore choose the sequence u ln (x, y) so 
that it converges at P L to this limit U 1 (x 1 ^ y^). Since, moreover, the func- 
tions u ln (x, y) are uniformly bounded their values at P 2 have at least one 
limit point U 2 (x 2 ,y 2 )'., we therefore select from the infinite sequence 
u\n ( x > y} a second infinite sequence u 2n (x, y) which converges at P 2 to 
U 2 (x< 2 , y 2 )- These functions u 2n (x,y) are uniformly bounded and their 
values at P 3 have at least one limit point / 3 (x 3 , ?/ 3 ), we therefore select 
from the sequence u 2n (x, y) an infinite subsequence u% n (x, y) which con- 
verges at P 3 to C7 3 (x 3 , ?/ 3 ), and so on. 

We now consider the sequence u u (x, ?/), u 22 (x, y), ^33 (^, y), Since 

the functions are all equicontinuous we have 

I ^m (*',</') - u mm (x,y) | < Je 
for any two points P and P' whose distance PP' is not greater than S. 

Next let 2~ q be less than 8 and let r be such that the set P 19 P 2 , ... P r 
contains all the points of R for which q has a selected value satisfying this 
inequality, then a number N can be chosen such that for m > N 

\U tt (Xs,ys) ~ ttmmfon!/*) I < !* 

for all values of Z greater than m and for all points (x s , y s ) for which q has 
the selected value. This number ^V should, in fact, be chosen so that the 
sequences u mm (x, y) converge for all these points P s . These points P 8 
form a portion of a lattice of side 2~ Q and so there is at least one of these 
points P' within a distance 8 from z. We thus have the additional in- 
equalities 

I u u (x,y) - Uu (x',y f ) | < K 

I u tt (#', y') - u mm (x f , y') | < $, 
KU (x, y) - u mm (x, y) | < . 

This inequality holds for all points P in R and proves that the diagonal 

sequence converges uniformly in R to a continuous limit function u (x, y). 

There is a similar theorem for sequences of functions of any number of 

variables, and for infinite sets of functions which are not denumerable. 



Equicontinuity 

In the case of a sequence of functions / x (2), f 2 (z), ... of a complex 
variable z = x -f iy there is equicontinuity when 

I/. (*')-/. (3) I < 4* 

for any pair of values z, z' for which | z' z \ < S, S, as before, being 
independent of s and of the position of z in the region E. A sufficient 
condition for equicontinuity, due to Arzela*, is that 



Z'-Z < 



for all functions f s (z) of the set and for all pairs of points z, z' of the domain, 
M l being a number independent of s, z and z' '. We need in fact only take 
SJ^j = ^e to obtain the desired inequality. 

In the particular case when each function / s (z) possesses a derivative 
it is sufficient for equicontinuity that |// (z) | < M 2 , where M 2 is inde- 
pendent of s and z. The result follows from the formula for the remainder 
in Taylor's theorem. 

Montel| has shown that if a family of functions f s (z) is uniformly 
bounded in a region R it is equicontinuous in any region R' interior to R. 

Suppose, in fact, that | /, (z) | < M for any point z in R and for any 
f unction f s (z) of the family, the suffix s being used simply a.s a distinguishing 
mark and not as a representative integer. 

Let D be a domain bounded by a simple rectifiable curve G and such 
that R contains D while D contains R'. We then have for any point 
within R' 

,, ,w If f* (z)dz 



Therefor* 



where I is the length of C and h is the lower bound of the distance between 
a point of G and a point of R' . This inequality shows that the functions 
f s (z) are equicontinuous in R' '. 

Now if f 8 (z) = u s (x, y) -f- iv 8 (x, y) where u a and v s are real, the func- 
tions u s , v s are likewise equicontinuous and uniformly bounded in R' . We 
can then select from the set u a a sequence u u , w 22 , u^ , . . . which converges 
uniformly to a function u (x, y) which is continuous in R'. Also from the 
associated sequence v n , v 22 , v& , . . . we can select an infinite subsequence 
v aa , v bb , v ce , ... which converges uniformly in R' to a continuous function 
v (x, y). The series 

fa* (*) + [/*> (Z) - faa (*)] + L/cc (*) ~ / (*)] + ... 

then converges uniformly to u (x, y) + fy (a;, y), which we shall denote by 

* Mem. delta R. Ace. di Bologna (5), t. vm. 
t Annales de Vtfcole Normale (3), t. xxiv, p. 233 (1907). 
B 19 



290 Conformal Representation 

the symbol / (z). On account of the uniform convergence the function 
/ (z) is, by Weierstrass' theorem, an analytic function of z in J?'. Indeed if 
/, (n) (z) denotes the nth derivative of any function f 8 (z) of our set and C' 
is any rectifiable simple closed curve contained within J?', we have by 
Cauchy's theorem and the property of uniform convergence 

faa (n) (z) + [fn (n) (z)~faa (n) (*)]+ 

n\ 



Since/ () is continuous in J?' and on C' the integral on the right represents 
an analytic function. When'n = this tells us that the sequence 

/oa ()>/ (*), 

converges to an analytic function which, of course, is / (z). When n ^ the 
relation tells us that the sequence / aa (n) (z), f bb (n) (z), ... converges to/ (n) (z). 
We may conclude from Cauchy's expression for f s (n) (z) as a contour 
integral that/ s (n) (z) is uniformly bounded in any region containing R r and 
contained in R. It then follows that the set/ s (n) (z) is equicontinuous in R . 
Hence from the sequence f aa (z), f bb (z),f cc (z), ... we can select an infinite 
sequence f aa (z), f ftft (z),/ yy (z), ... such that /' (z),f ftft r (z),/y/ (z), ... con- 
verges uniformly in R' to an analytic function which can be no other 
than /' (z). At the same time the sequence f aa (z),/^ (z), ... converges uni- 
formly to/(z). This process may be repeated any number of times so as 
to give a partial sequence of functions converging uniformly to / (z) and 
having the property that the associated sequences of derivatives up to an 
assigned order n converge uniformly to the corresponding derivatives of 



We now consider a sequence of contours C l) (7 2 , ... C n having for 
limit the contour C Q which bounds R, the contours C l9 (7 2 , ... C n bounding 
domains D l9 D% 9 ..., each of which contains the preceding and has .R as 
limit. From our set of f unctions f s (z) we can select a sequence f sl (z) which 
converges uniformly in D l towards a limit function, from* the sequence 
/i ( z ) we can cun * a new sequence f 82 (z) which converges uniformly in D 2 
to a limit function and so on. The diagonal sequence / n (z),/ 22 (z), ... then 
converges uniformly throughout the open region .R to a limit function. 

Hence we have Montel's theorem that an infinite set of uniformly 
bounded analytic functions admits at least one continuous limit function, 
both boundedness and continuity being understood to refer to the open 
region R in which the functions are defined to be analytic. 

For further developments relating to this important theorem reference 
must be made to Montel's paper. For the case of functions of a real variable 



Mapping of an Open Region 291 

A. Roussel* has recently invented a new method. The selection theorem 
has been extended by Montel to functions of bounded variation. 

4-46. Mapping of an open region. Let B be a simply connected bounded 
region which contains the origin and has at least two boundary points. 
Let S be the set of analytic functions f s (z) which are uniform, regular, 
smooth and bounded in R and for which 

/ s (0) = o, /;(0)=i, \f.(z)\<M. 

Let U s be the upper limit of | f s (z) \ in R and let p be the lower limit of all 
the quantities U s . There is then a sequence f r (z) of the functions/, (z) for 
which U r -> p. Since, moreover, this sequence is uniformly bounded, we 
can apply the selection theorem and construct an infinite subsequence 
which converges uniformly to a limit function / (z) in any closed partial 
region R' or R. This function / (z) is a regular analytic function in R and 
satisfies the conditions/ (0) = O,/' (0) = 1. Being a uniform limit function 
of a sequence of smooth mapping functions it is smooth in R and its U is p. 

The function / (z) thus maps R on a region T which lies in the circle 
with centre O and radius p. If T does not completely fill the circle, there 
will be a value re**, with r < p, which is not assumed by our function / (z) 
in R. We shall now show that this is impossible and that consequently T 
does fill the circle. 

Let r = a 2 p, then a < 1 and if we write 

f (z] - 2a 0( >i* J 7 (*) -JL 
h(Z} ~ l + a* pe av(z)~- 1' 

r / v-,0 /(z) - tfpe* 

where [f (* ] --zfT N ~ ^ > 

L v /J a 2 / (z) pe*v 

v (0) = a, 

we have / (0) = 0, /</ (0) = 1 and the function / (z) is uniform, regular, 
smooth and bounded in R. 

Now let U Q be the upper limit of / (z) in R. We may find an inequality 
satisfied by this quantity by observing that 

v (z) a 
av (z) 1 

is of the form ri/r 2 , where r ly r 2 are the distances of the point v (z) from the 
points a, I/a respectively which are inverse points with respect to the 
circle | z \ = 1. 

On the other hand, 

a 2 | v(z) | 2 = P!//^, 

where p l9 p 2 are the distances of the point f (z) from the points a^pe**, 
a" 2 pe^ which are inverse points with respect to the circle | z \ p. Now 

* Journ. de Math. (9), t. v, p. 395 (1926). See also Bull, des Sciences Math. t. LKL, p. 232 (1928). 

19-2 



292 Conformal Representation 

the point / (z) lies either on this circle or within it and so p!/p 2 has a value 
which is constant either on | z \ = p or on a circle within | z \ = p and with 
the same pair of inverse points. This constant for a circle with this pair of 
inverse points has its greatest value for the circle | z \ p if circles lying 
outside this circle are excluded. This greatest value is, moreover, 

^^ - ^ 

p - a~ 2 p 

Hence we have the inequality | v (z) | 2 < l. By a similar argument we 
conclude that r x /r 2 has its greatest value when the point v (z) is at some 
place on the circle | z \ = 1 and this value is a. Hence 

v (z) a 
av (z) - 1 

and so U (} < p. We have thus found a function for which C7 < p, and this 
is incompatible with the definition of p as the lower limit of the quantities 
U K . The region T must then completely fill the circle of radius p and so the 
function/ (z) maps E on this circle. The radius p is consequently called the 
radius of the region R. 

This analysis, which is due to L. Fejcr and P. Riesz, is taken from a 
paper by T. Rado*. The analysis has been carried further by G. Julia| who 
first selects from the functions f s (z) the polynomials p^ (n) (z) of degree n. 
Among these polynomials there is one polynomial p (n} (z) whose maximum 
modulus has a minimum value m n . It is clear that m n > p. Julia specifies 
a type of region R for which the sequence p (n) (z) possesses a limit function 
/ (z) mapping the region R on the circle of radius p. 

4-51. Conformal representation and the Green's function. Consider in 
the .T?/-plane a region A which is simply connected and which contains the 
origin of co-ordinates. We shall assume that the boundary of A is smooth 
bit by bit. We write 



for the Green's function associated with the origin as view-point, r being 
short for (a: 2 -f ?/ 2 )4. Let us write H (0, 0) = log r log (!//>), then r is the 
capacity constant or constant of Robin J. Now let 

Z= <f>(z) = z -f c 2 z + c 3 z 3 -|- ... 

be the uniquely determined function which maps the interior of a circle 
| Z | < p on A in such a manner that 

<f> (0) = 0, c' (0) = 1. 

* Szeged Ada, 1. 1, p. 240 (1923). 
| Complex Rendus, t. CLXXXIIT, p. 10 (1926). 

J The boundafy of A may also be taken to be a closed Jordan curve, in which case r is the 
transfmite diameter. 



Relation to the Green's Function 293 

It will be shown that the Green's function G (x, y) is 
(x, y) = log (p/r), f =|^( 2 )|. 

Bieberbach* has proved a theorem relating to the area of the region A 
which is expressed by the inequality area > Tip 2 . This means that among 
all regions A for which H (0, 0) has a prescribed value the circle possesses 
the smallest area. 

For the theorem relating to the Green's function we may, with ad- 
vantage, adopt a more general standpoint. Let us suppose that the 
transformation w = / (z) maps the area A on the interior of a unit circle 
in the w-plane in such a way that to each point of the circle there corre- 
sponds only one point of the area A and vice versa. Let the centre of the 
circle correspond to the point z of the area A , then z is a simple root of 
the equation / (z ) == and/ (z) = has no other root in the interior of A. 
This is true also for the boundary if it is known that there is a (I, 1) 
correspondence between the points of the unit circle and the points of the 
boundary of A . We may therefore write 

'fW-k-zJe*, 

where the function p (z) is analytic in A. 

Putting p (z) = P + iQ, z Z Q = re 1 *, where P, Q, r and 6 are all real, 
we have 

w=f(z) = exp {log r + P + i (Q + 9)}. 

Now, by hypothesis, the boundary of A maps into the boundary of the 
unit circle, therefore log r + P must be zero on the boundary of A . This 
means that log r -f- P is a potential function which is infinite like log r at 
the point (x , ?/ () ), is zero on the boimdary of A and is regular inside A 
except at (# , y Q ). This potential has just the properties of the function 
G (x, y\ X Q , 3/ ), where G (x, y, x , y ) is the Green's function for the area A 
when (x , y ) is taken as view-point, consequently the problem of the 
conformal mapping of A on the unit circle is closely related to that of 
finding the function G. 

.Writing a = Q -f- 0, < a < 2-rr, we have on the boundary of the circle 

dw = ie l *dcr, 

while on the boundary of A 

dz= \dz\ e*+, 

where ifj is the angle which the tangent makes with the real axis. Since 
dzfdw is neither zero nor infinite, the function 

dz\ 
dw) 



i log ( i 



L. Bieberbach, Rend. Palermo, vol. xxxvm, p. 98 (1914). This theorem is discussed in 4-91. 



294 Conformal Representation 

is analytic within the circle and its real part takes the value ^ a on the 
boundary of the circle. On the other hand, the function 

F (w) = - i log [- i (1 - w) 2 dz/dw] 

is analytic within the circle and its real part takes the value iff on the 
boundary. 

If is a known function of a on the boundary of A, Schwarz's formula 
gives , 





where k is an arbitrary constant. The preceding formula then gives z by 
means of the equation ^ M ^ 

_ <? n I _ _ 

(O - V I -j~ ~\~f) 

) W >(1 - w)* 

The relation between </r and a is partly known \fhen the boundary of 
A is made up of segments of straight lines but in the general case i/j is an 
unknown function of o- and the present analysis gives only a functional 
equation for the determination of 0. 

To see this we suppose that on the circumference of the circle 

F = i/j 4- ;</> 

where <f> and i/j are real, then 

| dz | = -J cosec 2 - | da | e~^, 

2i 

and the curvature of the boundary of A is 



and may be regarded as a known function of j/r, say (7 (i/r). Making use of 
the relation between cf> and i/j of 3' 33 

* (a) = ^ ~ 27T r ^' (a<>) iog [^ c sec2 a ?~] da ' 

where 6 is a constant, we obtain the functional equation* 



4(7(0) sin^ 
where (a) is defined by the foregoing equation. 

EXAMPLE 
Prove that 



[K. Lowner.] 

4-61. Scliwarz's lemma. It was remarked that a Taylor expansion for 
/ (w) in powers of w w is not required for points w on the real axis, 

* T. Levi Civita, Rend. Palermo, vol. xxni, p. 33 (1907); H. Villat, Annaies de rtfcole Normale, 
t. xxvm, p. 284 (1911); U. Cisotti, Idromeccamca piana, Milan, p. 50 (1921). 



Schwarz^s Lemma 295 

but when f(w) is real* for real values of w belonging to a finite interval, 
Schwarz has shown that it is possible to make an analytical continuation 
of / (w) into a region for which v is negative. Let us consider an area S 
bounded by a curve ACB of which the portion A B is on the line v = 
within the interval just mentioned. 

Let S' be the image of S in the line v = and let the value otf(w) for 
a point w' of S' be defined as follows. We write 

w = u + iv 9 w' u iv, 



where u, v, f, 77 are all real. The function f (w) being now defined within 
the region S + S' we write 

g (w, ) = 1/27T* (w - 
and consider the two integrals 



= 9 



f = f (7 (">, 

Js' 



taken round the boundaries of S and S' Since / (w) is analytic within 
both S and $' we have 

/=/()> /' = when lies within S, 
7=0, /'=/() when lies within S'. 
Hence in either case 7 + /'=/() and so 

/()=( <7(t0,)/(MOdt0, 



for the two integrals along the line ^4 J3 are taken in opposite directions and 
so cancel each other. 

Now the integral in this equation can be expanded in a Taylor series 
of ascending powers of - for any point within the area S -f S' 
whether is on the real axis or not. The integral in fact represents a 
function which is analytic within the area S f S' and can be used to define 
/() within S + /S". In. this case, when is on AB, f () can be expanded 
in a power series of the foregoing type and the coefficients in this series, 
being of type 



are all real. 

* It is assumed here that / (w) has a definite finite real integrable value for these real values 
of w. In a recent paper, Bull, des Sciences Math. t. LIT, p. 289 (1928), G. Valiron has given an 
extension of Schwarz's lemma in which it is simply assumed that the imaginary part irj of f (w) 
tends uniformly to zero as v -> 0. If, then, the function / (w) is holomorphic in the semicircle 
| w \ < R, v > 0, it is holomorphic in the whole of the circle | w \ < R. 



296 Conformal Representation 

Let us now use z to denote the value of z corresponding to this value 
of w. The equation 

z - z, +-f(w) -/(U = a (w - ) + b (w - )* + c (w - &) 4- ... 
can be solved for w by the reversion of series if a ^ 0, and the series 
thus obtained is of type 

w- = A(z-z ) + B(z- 2o ) 2 + C (z - z,,) 3 + ... , 

where the coefficients A, B, C are all real. The exceptional case a 
occurs only when the correspondence between w and z at the point 
ceases to be uniform. 

4-62. The mapping function for a polygon. Let us now consider an 
area A in the z-plane which is bounded by a contour formed of straight 
portions L 19 L 2 , ... L n . Let z denote a point on one of the lines L and 
let tin be the angle which this line makes with the real axis, also let W Q be 
the value of w corresponding to z. 

It is easily seen that the function 

/ (w) - (z - z ) e-*- 

has the properties of a mapping function for points z within A, and 
consequently also for the corresponding region in the w-plane; it is real 
when the point z is on the line L in the neighbourhood of z and changes 
sign as z passes through the value z ; consequently, when considered as 
a function of w it is real on the real axis and changes sign as w passes 
through the value w . Schwarz's lemma may, then, be applied to this 
function to define its continuation across the real axis and it is thus seen 
that we may write 

e -ih (z _ 2() ) =(20- WQ ) P ( W - W ), 

where P (w W Q ) denotes a power series of positive integral powers of 
w WQ including a constant term which is not zero. From this equation 
it follows that in the neighbourhood of the point w 



where P (w w ) is real when w and w are real. 

Taking logarithms and differentiating again, we see that the function 



= * (log &\ 
dw\ * dw) 



is real and finite in the neighbourhood of w = W Q . 

Next, let z l denote the point of intersection of two consecutive lines 
L, L' ', intersecting at an angle 0:77; the argument of z l z varies from hrr 
to fnr TT as the point z passes from the line L to L' through the point of 
intersection (Fig. 19). Hence the function 

1 
J= [(! - z)e~ lh *] a 




Mapping Function for a Polygon 297 

is real and positive on L and negative on L' . Moreover, it has the required 
properties within A , and when considered as a _, 
function of w it has the required mapping pro- 
perties in the region corresponding to A and 
is real on the real axis. By Schwarz's lemma 
we may continue this function across the 
real axis and may write for points w in the 
neighbourhood of w 1 , 

J = (w - Wi) PI (w - Wi), 

where P 1 (w w) is a power series with real 
coefficients and with a constant term which is 
not zero. This equation gives 

z z l = e* hir (w Wi) a P 2 (w w^, 

where P 2 (w w) is another power series with real coefficients. This 
equation indicates that for points in the neighbourhood of w l 

dZ .. . x 1 T> / x 

^ = *<- *>-* p a (-a 

where P 3 (w w) is a power series with real coefficients. Taking logarithms 
and differentiating we find that 

n / x d /, dz\ a 1 m , . 

F (w) = j- ( log j - ) = h T (w ~ WJ, 

dw \ dw) w w^ 

where T (w wj is a power series with real coefficients. The function 

F ( - a ~ l 
w Wi 

is thus analytic in the neighbourhood of w w l . 

For a point z 2 on the boundary of A which corresponds to ^v we have 
(if z 2 is not a corner of the polygon) 

2 w w 2 

Therefore -T- = lp\~ 

dw w 2 ^ \w. 



d A dz\ 2 1 /IN 

= j- ( log 3- ) = ---- h s P! I - , 
d?/; \ & dw;/ te; w; 2 / \w;/ 



where p (l/w) is a power series. The expansion for z z 2 may, indeed, be 
obtained by mapping the half plane w into itself by means of the sub- 
stitution w \IW-L , and by then using the result already obtained for 
an ordinary point z on L. 

The function F (w) is real for all real values of w, as the foregoing 
investigation shows, is analytic in the whole of the upper half of the 
w-plane and is real on the real axis, the fact that it is analytic being a 



298 Conformal Representation 

consequence of the supposition that the inverse function z g (w) is 
analytic in the upper half of the w-plane. Applying Schwarz's lemma we 
may continue this function F (w) across the real axis and define it analytically 
within the whole of the w-plane, the points on the real axis which are 
poles of F (w) being excluded. 

When | w \ is large | F (w) \ is negligibly small, as is seen from the 
expansion in powers of ljw\ moreover, F (w) has only simple poles corre- 
sponding to the vertices of the polygon A and these are finite in number. 
Hence, since F (w) outside these poles is a uniform analytic function for 
the whole w-plane, it must be a rational function. 

Let a, 6, c, ... 1 be the values of w corresponding to the vertices of the 
polygon and let arr, /?TT, ... XTT be the interior angles at these vertices, then 

/ x ^ a I d , , dz 

F (w) = 2 = -=- log -, , 

w a aw & aw 

and there is a condition 

S (a - 1) = - 2, 

which must be introduced because t}ie sum of the interior angles of a 
closed polygon with n vertices is equal to (n 2) TT. 
Integrating the differential equation for z we obtain 

z = c( (w - a)"- 1 (w - &V 3 - 1 ... (w - I)*" 1 dw + C", 

where C and C" are Arbitrary constants. By displacing the area A without 
changing its form or size but perhaps changing its orientation we can 
reduce the equation to the form 



= K 



\(w- a)*- 1 (w - by-* ...(w- I}*- 1 dw, 



where K is a constant. This is the celebrated formula of Schwarz and 
Christoffel* If one of the angular points with interior angle fin corresponds 
to an infinite value of w, the number of factors in the integrand is n 1 
instead of n, and the equation 

S (a - 1) = - 2 
may be written in the form 

S (a - 1) = - 1 - p, 

where now the summation extends to the n 1 values of a which appear 
in the integral. 

Since we can choose arbitrarily the values of w corresponding to three 
vertices of the polygon, there are still n 3 constants besides C and C' 
to be determined when the polygon is given. In the case of the triangle 
there is no difficulty. We can choose a, 6 and c arbitrarily; a, ]8 and y are 
known from the angles of the triangle and by varying K we can change 
the size of the triangle until the desired size is obtained. 

* E. B. Christoffel, Annali di Mat. (2), 1. 1, p. 95 (1867); t. iv, p. 1 (1871); Get. Werke, Bd. I, 
S. 265. H. A. Schwarz, Journ.filr Math. Bd. LXX, S. 105 (1869); Ges. Abh. Bd. n, S. 65. 



Mapping of a Triangle 299 

An interesting example of the conformal representation of a triangle 
with one corner at infinity is furnished by the equation 

z z = \ f (>s) ds, where / (6*) = (2a/7r) (1 s 2 )%/s, w = u ~\- iv. 
When w lies between 1 and oo we have 



= ft -f ic, say, 

where ft is a constant and c varies from to oo. Thus, the portion w > 1 
of the real axis corresponds to a line parallel to the axis of y. 
Again, if < w < 1, we may write 



i 

- ft - d, 

where d varies from to 06. The portion < w < 1 of the real axis corre- 
sponds, then, to a line parallel to the axis of x and extending from 

z --= z (i -f- ft to oo. 

When 1 < w < we may write 

r - 1 no 

*-*o= f(*)ds+ f(s)ds 
Ji J -i 

= ft' 4- d', 

and so the corresponding line in the z-plane extends from oo to ft' H- z 
and is parallel to the axis of x. When oo < w < 1, we have 



f ~ l (~ l 

- ZD = / (*) ds- \ f (s) ds 

Ji Jw 



- ft' - ic', 

where c' ranges from to oo, and so this part of the ^-axis corresponds 
to a line from ft' parallel to the i/-axis. The two lines parallel to the i/-axis 
can be shown to be portions of the same line separated by a gap. We have 
in fact 

b-b f =ff (s) ds - f "V (s) ds = f 1 / (s) ds, 

Ji Ji J -I 

where the integral is taken along the semicircle with the points 1, + 1 
as extremities of a diameter. On this semicircle we may put 



andso 



f i 

(7r/2a) (ft - ft') = i\ -* d0 [- 2i sin e^ 

J re 

fir / /3 /D\ 

-= i (1 - i) (cos - + i sin - J (sin 6)4 rf0 

- 2i f" cos ? (sin 9)% d0 = 2i^2T (J) T (f ) = i. 



300 Conformal Representation 

The figure in the z-plane is thus of the type shown in Fig. 20. To solve 
an electrical problem with the aid of this transformation we put d> = ie^ x , 
where % is the complex potential $ -f ifi. This transformation maps the 
half w-plane for which v > on a strip of the ^-plane lying between the 
lines <f> il: 77. 

Performing the integration we find that 



log ?-- - 2 V2 - 2 log 



where 



Fig. 20. 



-C, 



1-1 



Fig. 21. 



When the real part of w is large and negative the chief part of the 
expression for z is 



a 



log (r - 1) = - log (1 + & + ... - 1) = log 2 - - 



77 



77 



This gives a field that is approximately uniform. On the other hand, 
when the real part of w is large and positive, the chief part of the expression 
for z is 



and we may thus get an idea of the nature of the field at a point outside 
the gap and at some distance from its surfaces. These results are of some 
interest in the theory of the dynamo. Another interesting example, in 
which the polygon is originally of the form shown in Fig. 21, gives edge 
corrections for condensers*. 

Assigning values of w to the corners in the manner indicated, the 
transformation is of type 

\ 1 [( w + ^ ( w - b) dw 
CaC^Ce)-* V ---- -'-+-- ' 
J(w 0,1)* (w - 



n 

z - G 



-- ---- 

(w -f a 2 ) * 



r . . 

[(q - w) ... (c 6 + 



* J. J. Thomson, Recent Researches in Electricity and Magnetism, 1893; Maxwell, Electricity 
and Magnetism, French translation by Potier, ch. n, Appendix; J. G. Coffin, Proc. Amer. Acad. 
of Arts and Sciences, vol. xxxix, p. 415 (1903). 



Edge Correction for a Condenser 301 

Making a 8 -> 0, c 8 -> oo we finally obtain 



where G and 6 are constants to be determined. 
Integrating, we find that 

z = C [w + I (w - 6) 2 - 6 log (- w) + F], 

where F is a constant of integration. Since z = when w? = 1, we have 

F= 1- |(1 + 6) 2 . 

When ^ is small and positive, the imaginary part of z must be ^7?, and 
the real part must be negative. Since the argument of w in both con- 
ditions are satisfied by taking C = h/bir, therefore 

_ h 

Assuming that the potential </> is zero on A A and equal to V on BB 1 , 
we may write 

^ = i<^ = (log WJ ITT). 

7T 



Fig. 22. 

The charge per unit length on BB' from the edge (w = 6) to a point 
P (?# = s) so far from B that the surface density is uniform is 

1 V 

q== ~~ ^ p ~ ^ = ~ log (<S / 6)l 



47T 
Now when 5 is very small and positive, z = x -f i&, and so 

x + iA = _ --- (i - 26 - 2i&7r - 26 log 5). 



Therefore log (s/b) = irx/h + 1/26 - 1 - log 6, 

F TTX 1 - 26 



, 
and so 8s= 

When b = 1 we have the well-known result 



, /] 
_log6j. 



in which it must be remembered that x is negative. 



302 Conformal Representation 

When w is very small and negative, z x, and so 



2 Io 

V P 

W>, LTT 

V h 



, rl , , V (h \ 
VV hen b \ q . -. I x } , 

47T//, \277 / 

A f/ ( h \ 

6-00 r/ = - - - x }. 

A ~ L \7T ] 



Since ?^ -- 6 at the point B, the value of 2 for this point is 
3 = - 5 f- [1 - 6 2 - 26 log 6 - 26i7r], 

2u7T 

and so the upper plate projects a distance d beyond the lower one, where 



Many important electrostatic problems relating to condensers are solved 
by means of conforinal representation in an admirable paper by A. E. H. 
Love*. The problem of the parallel plate condenser is treated for planes 
of unequal breadth and for planes of equal breadth arranged asymmetrically. 
The formulae involve elliptic functions. The hydrodynamical problems 
relating to two parallel planes, when the motion is discontinuous, are 
treated in a paper by E. G. C. Poolef. 

Some applications of conformal representation to problems relating to 
gratings are given in a paper by H. W. Richmond^. The general problem 
of the conformal mapping of a plane with two rectilinear or two circular 
slits has been discussed recently by J. Hodgkinson and E. G. C. Poole. 

4-63. The mapping function for a rectangle. When n 4 and 
a ^ p ^ y -= 8 = ^, the polygon is a rectangle and z is represented by an 
elliptic integral which can be reduced to the normal form 

z = H f dt [(1 - t*) (1 - & 2 * 2 )]-i 
Jo 

by a transformation of type 

w(Ct + D) - At + B. ...... (A) 

If, in fact, the integral is 

dw [(w p) (w q) (w r) (w s)]~% 9 

we have (Ct -f D) (w - p) - (A - Cp) t -f B - Dp, 

(Ct -h D 2 ) dw - (AD - BC) dt, 



* Proc. London Math. Soc. (2), vol. xxn, p. 337 (1924). f ^id. p. 425. 

J Ibid. p. 389. Ibid. vol. xxra, p. 396 (1925). 



Mapping of a Rectangle 303 

and so the transformation reduces the integral to the normal form if 

A - Cp = B - Dp, 
A - Cq = Dq - B, 
A - Cr = k (B - Dr), 
A - Gs = Ic (Ds - B). 
These equations give 

C (q - p) = 2B - D (p 4- g), 
C (s - r) = 2kB - kD (r + s), 
2A - (7 (p + <?) = JQ (q - p), 
2A - C (r + s) = kD (s - r), 
C[s-r- k(q-p)]= kD[p + q-r- s], 
C [r + s - (p + q)] - D [q - p + k (r - a)], 

2 (? _ p) (r _ 3) + k [(q _ p) 2 + (r _ 0)3 _ (p + ? __ f __ j )S] 

+ (? ~ P) ( r ~ s) = 0. 
This equation gives two values of k which are both real if 

[(? ~ P? + (r ~ ) 2 - (p + ? - r - s) 2 ] 2 > 4 (gr - ^)) 2 (r - s)*, 
that is, if 

[(? - P + r ~ s ) 2 ~ (P + ? ~ r - 5 ) 2 1 [(? ~ P ~ r + <*) 2 ~ (p + ? - r - 5) 2 ] > 0, 
or, if 4 (# s) (r p) (q r) (s p) > 0. 

Itr^p^>q^s this is evidently true and since the product of the two 
values of k is unity, we may conclude that one value of k is greater than 1 , 
the other less than 1. This latter value should be chosen for the transfor- 
mation. With this value 



consequently, the transformation (A) transforms the upper half of the 
w-plane into the upper half of the -plane. 

When the normal form of the integral is used the lengths of the sides 
of the rectangle are a and b respectively, where 



a - H dt [(1 - t 2 ) (1 - W)]-* - 2HK, 
J-i 

ri/fc 
6 = ff I dt [(1 - t*) (1 - 4V)]r* = HK', 

and where 4X and 2iK are the periods of the elliptic function sn u defined 
by the equation x = sn u, where 

u^ \ X dt[(l - t*) (1 - tV)]-. 
Jo 

With the aid of this function can be expressed in the form 

t - sn (z/H). 



304 Conformal Representation 

The modulus k may be calculated with the aid of Jacobi's well-known 

r(\ -fg'Ml + ) (1 f. 



formula 



in which </ = exp [ TrK'/K] = exp [ 2?r6/a]. 

When the region is of the type shown in Fig. 23 the internal angles of 
the polygon are 877/2 at four corners and ?r/2 at the other eight. The 
transformation is thus of the type 
2 - A [(w - cj (w - c 2 ) (w - c 3 ) (w - c 4 )]* [(w - fa) ... (w - p B )]~*dw + B. 

A particular transformation of this type is obtained by assigning 
positive values of w to corners of the polygon which lie above the axis of 
x and negative values of w to corners which lie below the axis of x, points 
which are images of each other in the axis of x being given parameters 
whose sum is zero. The transformation is now 



z = 






Fig. 23. Fig. 24. 

Making the parameters a x , a 2 tend to zero and the parameters 6 lf 6 2 
tend to infinity, the transformation becomes 



, (w* - 



B, 



and the interior of the polygon becomes a region which extends to infinity. 

To use this transformation for the solution of an electrical problem in 
which the two pole pieces in Fig. 24 are maintained at different poten- 
tials, we write* , = ; C M*, x = + ^, 

so as to map the half of the w-plane for which v > on the strip TT < <f) < TT. 
This will make w> = correspond to z = if B = 0, and the lower limit of 
the integral is i \/c. 

Writing ck = 1 we find that the lengths a and 6 in the figure are given 

by the equations re ^ 

26= <7c ~ a /(a), 

./i 5 

^ f 1 d$ , . ~ [~ l ds j. . . 
- 2*a = Cc - 2 / (5) - Cc -2 / (a), 

Jt*/c 5 JiJc s 

* Riemann-Weber, Differentialgleichungen der Physik, Bd. n, S. 304. 



Region outside a Polygon 305 

where f (s) = [(1 s 2 ) (1 k 2 s 2 )]*. These integrals are easily reduced to 
standard forms of elliptic integrals, thus 
c ds - c ds 



ds ** c ds 



Now if we put ksr = 1, the last integral becomes 

~ _ r (L~JL?!L^ 

~ Ji ~7F) 



and we eventually find that 

6 = Cc[2E f - (I- k*)K'], 
a- 2Cc[2E - (1 - i a ) JL]. 
26 2^' - (1 - A 8 ) #' 



Therefore 



a 2E- (I - P) K ' 



EXAMPLE 
Prove that if OABC is the rectangle with sides x = 0, x = K, y = 0, y = K' and 

^ -f it/t = log (an z), 
we have ^ = on OA, AB t BC; == rr/2 on CO. Prove also that if 

^ 4- itj, = log (en z), 

where (en z) 2 4- (sn z) 2 = 1, we have = on 0.4, 0(7; t/i = n/2 on J5^4, J5C. 
See GreenhilTs Elliptic Functions, ch. ix. 

4-64. Conformal mapping of the region outside a polygon. In order to 
map the region outside a polygon on the upper half of the w-plane, we may 
proceed in much the same way as before, but we must now use the external 
angles of the polygon and must consider the point in the w-plane which 
corresponds to points at infinity in the z-plane. Let us suppose that the 
w-plane is chosen so that this point is given by w = i, then there should 
be an equation of the form 

~ C C (w - i) 
where the coefficients C m are constants. This gives 

_? _ i Q i 

dw (w i) 2 l *"' 

d , dz 2 -~ 

log ~- = . -h P (w ^), 

aw aw w i 

where P (w i) is a power series in w i. 

Since -3 log -j is to be real it must be of the form 

dw & dw 

1 - S a -^~ l - 2 2 

dw dw w a w i w + i % 

B 20 



306 Conformal Representation 

Therefore 

w- a)"- 1 (w - 6)*- 1 ... (w - iy~ l (1 4- w*)~*dw + C', ...(I) 

where C and C' are arbitrary constants of integration. The relation between 
the indices a is now 

S (a - 1) = 2, 

for the sum of the exterior angles of a polygon with n vertices is (n 4- 2) TT. 
The region outside a polygon can be mapped on the exterior of a unit 
circle with the aid of a transformation of type 



- a)*- 1 (w - b)?- 1 ... w~ 2 dw, | a | = | 6 | = ... = 1, 

where, as before, 

S (a - 1) - 2. 

When the integrand is expanded in ascending powers of w~ l there will 
be a term of type w 1 which will, on integration, give rise to a logarithmic 
term unless the condition 

Sa(a- 1) = 
is satisfied. 

When the polygon has only two vertices and reduces to a rectilinear 
cut of finite length in the z-plane, we have a = ft = 2. The second condition 
may be satisfied by assigning the values w 1 to the ends of the cut. 
The transformation is now 



= H f (w 2 - l)w- 2 dw, 



and the length of the cut evidently depends on the value of H. Taking 
H for simplicity, the transformation becomes 

2z = w 4- w~ l . 
This is the transformation discussed in 4-73. 

The general theorem (I) indicates that the regibn outside a straight 
cut may be mapped on the upper half of the w-plane by means of the 
transformation 

ft f 1 - w 2 , 2w 

ty O I fJIH 

Z ^ I 71i , oTo w -i ," o 

J (1 4- w 2 ) 2 1 4- w 2 

The region outside a cut in the form of a circular arc may be obtained 
from the region outside a straight cut by inversion. If the arc is taken to 
be that part of the circle z = ie* ie , for which a < 6 < a, the trans- 
formation 

4-14- 2iw tan a 



z = 



w 2 4- 1 2iw tan a 



maps the region outside the arc on a half plane. 

Suppose that in the w-pjane there is an electric charge at the point 
w = i (sec a 4- tan a) = is, say, and that the real axis is a conductor. 



Semicircular Arc 307 

The corresponding charge in the z-plane will be at infinity and the circular 
arc will be a conductor which must be charged with a charge of the same 
amount but of opposite sign. The solution of the potential problem in the 
w-plane is evidently 

i , - 1 i w is 
y = -f ii/f = log --- . 

* V -T- "T 6 w _j_ ls 

mu- i.i_ /i \ - 1 + &~ x sin 

This gives ur = - is coth 



and finally 

X= log ^ cosec a {z 4- 1 4- (z 2 4- 2iz cos 2a 1)1} . 

The two-valued function (z 2 + 2^z cos 2a l)i may be regarded as 
one-valued in the region outside the cut and must be defined so that it is 
equal to i when z = and is of the form z i cos 2a when j z \ is very 
large. Changing the signs of <f> and x we have 

x = K [2 sin a cosh <f> sin -f sin 2 a sin 20], 
^ = _ jf [l -f- 2 sin a cosh </> cos 4- sin 2 a cos 20], 
where K~ l = 1 4- 2e~^ sin a cos -f e" 2 ^ sin 2 a. 



With the aid of these equations Bickley has drawn the equipotentials 
and lines of force for the case of a semicircular arc. The charge resides for 
the most part on the outer face, the surface density becoming infinite at 
the edges. The field appears to be approximately uniform ori the axis just 
above the centre of the circle. 

The field at a great distance from the circular arc is roughly that due 
to an equal charge at the point z = i cos 2 , for when x is large, the 
equation 

. 1 -f e* sin a . r , . ._ _ , 

z = i -.- ------ --. = i [1 -f e x sin a] [1 e~* sin a] ... 

1 4- e~ x sin a L J 

may be written in the form 

z 4- i cos a a = ie x sin a 4- negligible terms. 

This point, which may be called the " centre of charge," is the middle 
point of that portion of the central radius cut off by the chord and 
the arc. 

On the circular arc 

X = i0 and z == ie 2ld . 

Therefore sin (0 0) sin a = sin 9. 

The surface density is thus proportional to S cos 8 4- 1 on the convex 
face and to S cos 6 1 on the concave face, S denoting the quantity 

S = (sin 2 a - sin 2 0)"*. 
If E is the charge per unit length of a cylindrical conductor whose 



308 Conformal Representation 

cross-section is the circular arc and d is the diameter of the circle, the 
surface density a is given by Love's formula 

2-rrcrd = E | sec v \ (cosec a cos v), 

where sin v = tan cot a. 

The solution of the electrical problem of a conducting plate under the 
influence of a line charge parallel to the plate but not in its plane may be 
derived from the preceding analysis by inversion from a point on the 
unoccupied part of the circle. Let AB be the cross-section of the plate, 
/)'the foot of the perpendicular from O on AB, OC 1 the bisector of the 
angle AOB, then the surface density cr is given by Love's formula 

_ E OD cosec a cos v 
= 277 OP 2 "J cos ~v~\ ' 
where now sin v = cot a tan (P'OC'), 

cos v = cosec a ^-p - , 

A'P'C'B' being perpendicular to OC 1 (Fig. 25). Thus 
__ E OD OA' =f (A'P' . J5'P')_* 
07 ~ 277 OP 2 (A'P'.WP')* 

O 







Fig. 25. 

This is easily converted into the expression given in 3-81. 
The region outside a rectangle may be mapped on the interior of the 
unit circle in the -plane with the aid of the transformation 



z= f ds ( 1 - 2s 2 cos 2a + 
Ji 



while a transformation which maps the region outside the rectangle into 
the region outside the circle is obtained by using a minus sign in front of 
the integral. 

Let us use this transformation to determine the drag on a long thin 
rod of rectangular section which is moved slowly parallel to its length 
through a viscous liquid contained in a wide pipe of nearly circular section. 
We write log = iw = i (u -f iv), where v is the velocity at any point in 
the z-plane, then 



= 2* f (cos 2a - cos 2s)$ ds = 2 f (sin 2 a - sin 2 <$)* ds. 
Jo Jo 



Hydrodynamical Applications 309 

Putting sin s = sin a sin 0, this becomes 



- 2 f ?^lizJE^!)_f - 2 I* (1 - sin* si 
Jo (1 - sin 2 a sin 2 j3)* Jo 

- 2 f cos 2 a (1 - sin 2 a sin 2 j8)~i 
Jo 



At the corner A immediately to the right of the origin in the z-plane, 
we have 6 = |TT, and 

X A = 2E (k) - 2k' 2 K (k), 

where k = sin a and E (k), K (k) are the complete elliptic integrals to 
modulus k. The drag on the half side OA of the rectangle is proportional 
to W A) and since 

sin W A sin a sin (|TT) = sin a, 

we have W A = a. The drag on the side OA is thus equal to (a/2?r) times the 
drag of the whole rectangle. [C. H. Lees, Proc. Roy*. Soc. A, vol. xcn, 
p. 144 (1916).] 

EXAMPLE 

A line charge Q at the origin is partly shielded by a cylindrical shell of no radial 
thickness having the line charge for its axis, the trace of the shell on the #y-plane being that 
part of the circular arc z = ae 2lB for which TT < 2co < 20 < 2o> < TT. Prove that the 
potential <f> is given by the formula 

( z ~ a ) cos2 "> -f- (z -f a) sin 2 a> -f R 



a) 8n _ (z _) 

where J2 denotes that branch of the radical [z 2 2az cos 2aj -f a 2 ]*, whose real part is 
positive when the point z is external to the circle. 

The surface density a of the induced charge at a point on the charged arc is 

a = _ Q { S ec a> (tan 2 o> - tan 2 0)~* l}/47ra, 

the upper sign corresponding to the density on the concave side, the lower sign to the density 
on the convex side. The latter is zero when 2o> = -n-, that is, when the circle closes. 

[Chester Snow, Scientific Papers of the Bureau of Standards, No. 642 (1926).] 

4-71. Applications of conformal representation in hydrodynamics. 
Consider the two-dimensional flow round an airplane wing whose span is 
so great that the hypothesis of two-dimensional flow is useful. Let u, v 
be the component velocities, p the pressure, p the density, L the lift per unit 
length of span, D the drag per unit length and M the moment about the 
origin of co-ordinates, this moment being also per unit length. These 
quantities may be calculated from the flux of momentum across a very 
large contour C which completely surrounds the aerofoil. In fact, if /, ra 
are the direction cosines of the normal to the element ds, we have 

L -f iD = p \ (v -f iu) (ul -f vm) ds \ p (m -f il) ds, 

M = p \ (xv yu) (ul + vm) ds \ p (xm yl) ds\ 



310 Conformal Representation 

the sign of M is such that a diving couple is regarded as positive. The 
equations may be rewritten in the form 

L + iD = l/o j (v -f fw) 2 dfe - I [p + Jp (^ 2 -f v 2 )] (m -f fZ) ds, 
M = \p\ [(u 2 - v 2 ) (mx 4- ly) -f 2tw (my - Zx)] efo 



f (ly ~ mx)[p 



-f 



where 2 = x -f- iy. Now when the motion is irrotational outside the aerofoil 
the quantity p -f \p (u 2 -f v 2 ) is constant, also 

(m -f iZ) rfs = 0, (ly mx) ds = 0, 

hence L + iZ> = lp \ (v 4- ft*) 2 rfz. 

Taking the contour to be a circle of radius r, we have 

(y 2 - x 2 )] ds/r 



f [^ 2 - ^ 2 - 2fwv] [x 2 - y 2 + 2ixy] ds/ir 

(^ -h iu) 2 zdz, 

where the symbol R is used to denote the real part of the expression which 
follows it. These are the formulae o^Blasius* but the analysis is merely 
a development of that given by Kutta and Joukowsky. 

The integrals may be evaluated with the aid of Cauchy's theory of 
residues by expanding v -f iu in the form 



When the region outside the aerofoil is mapped on the region outside 
the circle | z' \ a by a transformation of type 



the flow round the aerofoil may be made to correspond to a flow round 
the circle by using the same complex potential x i n each case. Now for 
our flow round the circle we may write 

X '- /v* tf 2 /?' 2 4- IK 

e a/z + 



Zeits.f. Math. u. Phys. Bd. Lvm, S. 90 (1909), Bd. LIX, S. 43 (1910). 



Region outside an Aerofoil 311 

where V, a and K are constants, therefore 

. dv . dv dz' 

v + iu = i -~ = i -= . - 

dz dz dz 

'e-*- We 



i/c KC a inc* U' . 
- -f s- --- ~ ~ 

27T2 



If 2' = ae lf * is the point of stagnation on the circle which maps into the 
trailing edge of the aerofoil, we have 

K = 2-jraU' sin (a - /?), 
U = nCTe, 
L + iZ> - KpU'ne-* = K pU = 2<napUU r sin (a - /?). 



4-72. jT/^e mapping of a wing profile, on a nearly circular curve. For 
the study of the flow of an inviscid incompressible fluid round an aerofoil 
of infinite span, it is \iseful to find a transformation which will map the 
region outside the aerofoil on the region outside a curve which is nearly 
circular. 

If the profile has a sharp point at the trailing edge at which the tangents 
to the upper and lower parts of the curve meet at an angle a, it is convenient 
to make use of a transformation of type 



Z + KC 

where cr = (2 K) TT. 

If a circle is drawn through the point c in the -plane so that it just 
encloses the point c, cutting the line ( c, c) in a point c -f d, say, where 
d is small, this circle will be mapped by the transformation into a wing- 
shaped curve in the z-plane. This curve closely surrounds the lune formed 
from two circular arcs meeting at an angle a at each of their points of 
junction, z == *c, z = KC. The curve actually passes through the point 
KC and has the same tangents there as the lune derived from a circle in 
the -plane which passes through the points ( c, c) and touches the former 
circle at the point c. 

If we start with the profile in the z-plane and wish to derive from it 
a nearly circular curve with the aid of a transformation of this type, the 
rule is to place the point KC at the trailing edge and the point KC inside 
the contour very close to the place where the curvature is greatest*. This 
rule works well for thin aerofoils, but it has been found by experience that 
by increasing the magnitude of d an aerofoil with a thick head may be 
obtained from a circle, and that the thickness of the middle portion of the 
aerofoil is governed partly by the value of cr. Hence in endeavouring to 

* F. Hohndorf, Zeits. f. ang. Math. u. Mech. Bd. vi, S. 265 (1926). 



312 Conformed Representation 

map a thick aerofoil on a nearly circular curve the point KC may be taken 
at an appreciable distance from the boundary. Another point to be noticed 
is that when a circle is transformed into an aerofoil by means of the 
transformation (A) the smaller the distance of the centre from the line 
( c, c) the smaller is the camber of the corresponding aerofoil and the more 
symmetric is the head. The point KC, moreover, lies very nearly on the line 
of symmetry. 

The actual transformation may be carried out graphically with the aid 
of two corresponding systems of circles indicated by the use of bipolar 
co-ordinates. The circles in one plane are the two mutually orthogonal 
coaxial systems having the points ( c, c) as common points and limiting 
points respectively; the corresponding circles in the other plane for two 
mutually orthogonal systems having the points ( c, c) as common points 
and limiting points respectively. This is the method recommended by 
Krm&n and Trefftz. Another construction recommended by Hohndorf 
depends upon the substitutions 



by which the transformation may be written in the form 



where 

The plan is to first consider the transformation from z to , given by 
the equation 

,1 C, ~~" C (Z ~~~ KC\ * 

r = t* or f = ( ) . 
-f- c \z 4- KC) 

This transformation may be performed graphically* by writing 

z = z + KC, = J + c, 
when the relation becomes 



When $ has been found its ^th root may be determined graphically 
and when this is multiplied by $ the value of r is obtained, and from this 
is easily derived. 

Hohndorf gives a table of values of rj corresponding to different 
angles a. When a -= 4, 77 = 89, when a = 8, 77 = 44, and when a = 10, 

* The transformation may also be performed graphically by writing it in the form 

-5('*?) 

when it is desired to pass from a figure in the (-plane to a corresponding figure in the 2-plane. 



Aerofoil of Small Thickness 313 

rj = 85. When the transformation (A) is expressed by means of series, 
the results are 

_, /c 2 - 1 c 2 (/c 4 - 5* 2 + 4) c 4 

+ > 



r l-/c 2 c 2 (4/c 4 - 5* 2 + l)c 4 

r ~ i __ __ v __ _ __ _ _ _i _ _i_ 

^ ^ 3 z 45z 3 ^"" 

4-73. Aerofoil of small thickness*. We have seen that the trans- 
formation 

z' = z + a 2 /z 

maps the circle (7 given by | z \ = a into a flat plate P' extending from 
z' = 2a to z' = 2a and back. On the other hand, if the A's are small 
quantities the transformation 

= s{l+ S ^n(a/2)"} 

n-O 

maps C into a curve F differing slightly from a circle, and if we then put 



F maps into a curve II' differing slightly from a flat plate. Now for a 
point on F 

= a(l + r)e ie , 

where 6 is a real angle and r a real quantity which is small ; therefore to the 
first order in r 

= 2a (cos + ir sin 0), 

and so ' = 2a cos 0, rj f = 2ar sin 0. 

For points on (7 and P' we may use a real angle co and write 

z = ae ia> , a?' = 2a cos co, y' == 0, 

then (1 4- r) e--) = 1 + S ^e-'-, 

n0 

i 

and since o> is small we have to the first order, with A n = B n + iC n , 

r = S (J5 n cos ^^6o -f <7 n sin 7^0), 
</> = 2 (C n cos nco -B n sin /io>). 
Hence by Fourier's theorem 

D r AJQ (* ^ c s n ^ 

7r.B n = r cos n0d6 = -- . w dO, 



* H. Jeffreys, Proc. Roy. Soc. A, vol. cxxi, p. 22 (1928). 



314 Conformal Representation 

Since sin and sin n0 are odd functions of 0, whilst cos nO is an *even 
function of 6, it appears that C n depends on the sums and B n on the 
differences of the values of rj' corresponding to angles #; thus the (7 n 's 
depend on the camber of the aerofoil, the J? n 's on its thickness. 

When 6 is small, that is, for points near the trailing edge of the aerofoil, 
we have approximately 

v) _ r sin __ 2r 
2a~~-~? ^ r~"cos = ~0 ' 

and when TT is a small quantity co we have 



Thus r vanishes at = because the slope of the section is finite there ; 
but at = TT the section and the axis meet at right angles at a point which 
may be called the leading edge. If the curvature at this point is l/R we 
have to a close approximation 

nr V 2 4a 2 r 2 sin 2 # _ . ., m . 9 

2jR = ,. -s- = :r~ ,- --- --, = 2ar 2 (1 cos 5) = 4ar 2 , 

^ + 2a 2a (1 4- cos 0) v ; 

consequently r is finite and equal to (R/2a)% at the leading edge. If at 
a great distance from the circle C the flow in the z-plane is represented 
approximately by a velocity U making an angle a with the axis of x, we 
have 

IK 
x = </, + ie/r = C7ze-* -f Ue^at/z + ^ log (z/a), 

where K is the circulation round the cylinder. Taking x to be the complex 
potential for a corresponding flow in the '-plane the component velocities 
(u 9 v) in this plane are given by the equation 



To determine K we make the velocity finite at the trailing edge where 
f ' is a maximum and = 0, r = 0, = a, -^ r = 0. 
Hence d^/dz is zero and so 

But when 0=0 



say, where j8 == S C n . 

Therefore /c = naU sin (a + 



Thin Aerofoil 315 

Now 



{1-2 '(n - 1M M (a/zf}"(l - a'/T 2 ) 

1 ' (1 + fl > + 4 1 o/g')/2"' , 

" 



+ IK 



Therefore by the Kutta-Joukowsky theorem the lift per unit span of 
the aerofoil is 



L - - = 477/>a (1 + 5 ) sin (a + jB) F 2 , 7(1 + J5 ) = J7, 

1 H~ -^0 

and the lift coefficient is 

# L = (L/4paV*) = 77 (1 + J? ) sin ( + )8). 

The thickness thus affects the lift through J3 , which is a positive 
constant for a given wing. 

The moment about 0, that is a point midway between the leading and 
trailing edges of the aerofoil, is equal to np times the real part of the 

coefficient of -f i'~ 2 in the expansion of (-= j . This coefficient is 



and so to the first order in the J3's and C"s the moment is M , where 
M - Z-n-pVW {C 2 cos 2a - (1 + B 2 ) sin 2a f 2B l cos a sin (a + ]8) 

4- 2^ sin a sin (a + ]3)}. 
The moment about the leading edge is 



where terms of orders a 2 , aB n> aC n have been retained, but terms of orders 
a 3 , a 2 J? n , a 2 C n , dropped. When squares and products of the JB's and C"s 
are neglected, the moment coefficient K M is 



= \K L (1 + B. + B,- 
The moment coefficient at zero lift is thus 



and is independent of the thickness to this order of approximation. The 
thickness, however, affects the coefficient of K L . 



316 Conformal Representation 

For further applications of conformal representation in hydrodynamics 
the reader is referred to H. Glauert's Aerofoil and Airscrew Theory (Cam- 
bridge, 1926) and to H. Villat's Lemons sur VHydrodynamique (Gauthier- 
Villars, Paris, 1929). 

4 81. Orthogonal polynomials associated with a given curve*. Let/ (z) 
be a function which is defined for points of the z-plane which lie on a closed 
continuous rectifiable curve C which is free from double points. If 

ds = I dz I , the integral r 

f(z)ds 

J C 

denotes as usual the limiting value 



lim 2 

m->co fl 

where z , z l , z 2 - 5 . . . represent successive points on C for which 

lim [Maximum value of | z v z v _\ \ for < v < m] 0, (z l = z m ), 

in > oo 

and denotes an arbitrary point of C which lies between z v ^ and z v . We 
have in particular / 

ds = l, 

Jc 

where I denotes the length of the curve C. We shall suppose now that the 
unit of length is chosen so that I = 1 . 

Using z to denote the complex quantity conjugate to z, we write 



D =sAoo = 



n . 



Let H mn denote the co-factor of h mn in the determinant D n , and let a n 
be a constant whose value will be determined later ; then, if 

P n (z) = a n (H 0n + zH ln + ... z n H nn ), 
it is easily seen that 

f P n (z) z v ds - a n (h Qv H 0n + h lt ,H ln -f ... h n H nn ) 
Jc 



- a n D n for v - n, 
P n (z) | 2 dz = a n a n H nn D n - a n 



* See a remarkable paper by Szego, Jtfa^. Zette. Bd. rx, S. 218 (1921). 



Orthogonal Polynomials 317 

The polynomials thus form an orthogonal system which is normalised 
by choosing a n , so that a n = a n = (A Ai-i)"*- 

If P m (z) denotes the complex quantity conjugate to P m (z), the ortho- 
gonal relations may be written in the form 



P n (z) P m (z) ds= 0, m*n 
Jc 

= 1, m = n. 

Let us now suppose that C is an analytic curve and that = y (z) is 
the function which maps the interior of C smoothly on the region | | < 1 
of the -plane in such a way that y (a) = 0, y (a) > 0. Since C is an 
analytic curve y (z) is also regular and smooth in a region enclosing the 
curve C. It is known, moreover, that there is one and only one function, 
z = g (), which is regular and smooth for | | < 1 and maps the interior 
of | | = 1 on the interior of the curve C. The derivatives of the functions 
y (z), g () are connected by the relation y (z) g' () = 1, where z and are 
associated points of the two planes. 

Our aim now is to show that 



o r z 

= lim ~-,J-r [K n (a,w)]*dw, 

n _>oo J^n V a > a ) J a 



where K n (a, z) = P (a) P (z) + ... P n (a) P n (z). 

We shall first of all prove an important property of the polynomial 
K n (a, z). 

Let a be arbitrary and G n (z) a polynomial of the nih degree with the 
property 

f \G n (z)\*ds = 1, 
Jc 

then the maximum value of | G n (a) \ 2 is K n (a, a), and this value is attained 
when G n (z) = eK n (a, z) [Jf n (a, a)]~~i, where e is an arbitrary constant 
such that | e | = 1. 
Let us write 

G n (z) = ^ P (z) + ^P! (z) + ... t n P n (z), 

where the coefficient t v is determined by Fourier's rule and is 



= f 

J 



We then have 

(z)\*ds= |* |*-f |^| 2 + ... \t n \\ 



= f 
J 



C 

G (a) = / P (a) + ^P! (a) + ... ^ n P n (a), 
and by Schwarz's inequality 



318 Conformal Representation 

The sign of equality can be used when 

t, = e!\W[K n (a,a)]-*-, 
that is, when O n (z) = eK n (a, z) [K n (a, a)]~i. 

When the point a is within the region bounded by C y and F (z) is any 
function which is regular and analytic in the closed inner realm of (7, we 
have the inequality 



where 8 is the least distance of the point a from the curve C. To prove 
this we remark that Cauchy's theorem gives 



and so | F (a) \ < ~ f ~- (z) ds < ~ [ \F (z) I da. 

' ' 2n Jc z a 277-8 Jp ' ' 

In the special case when F (z) = \G n (z)] 2 the inequality gives 



This is true for all polynomials G n (z), and so, in particular, 

27r8 
Since 8 is independent of n, this inequality establishes the convergence 

of the series 

K(a,a)= |P (a)| 2 + | P l (a) | 2 + ... 

for the case in which the point a lies in the region bounded by C. Again, 
we have the inequality 

\K n (a, 2) | 2 < [|P (a) || Po (z) | + | P l (a) \\ P,(z) \ + ...]< K n (a, a) K n (z,z), 
and if R n m (a, z) denotes the remainder 

Rn m (a, z) = P^TF) P n+l (z) 4- ... P n+m (a) P n+m (z), 
we have | R n m (a, z) | 2 < R n m (a, a) R n m (z, z). 

Since the series K (a, a) is convergent when a lies within C we can find 
a number N (a) such that if n > N (a) we have for all values of m 

| R n m (a, a) | < e, 

where e is a small positive quantity given in advance, hence if N is the 
greater of the two quantities N (a), N (z), we have for n > N 

This establishes the convergence of the series 



K (a, z) - KM P ft (z) + Pn(a) P, (z) + .... 



Series of Polynomials 319 

To prove that the series is uniformly convergent in any closed realm R 
lying entirely within C we note that the quantities K n (a, z) are uniformly 
bounded in the sense that 

| K n (a, z) | 2 < K n (a, a) K n (z, z) < 



The general selection theorem of 4-45 now tells us that from the 
sequence K n (a, z) we may select a partial sequence of functions which 
converges uniformly in R towards a limit function/ (z). Since, however, 
the sequence converges to K (a, z) this limit function/ (z) must be identical 
with K (a, z) and so the series which represents K (a, z) converges uniformly 
in R, a and z being points within R. 

We now consider the integral 



= [ 

J 



K n (a,z)-X{ 7 '(z)}l\*ds, 
c 

where A is a constant which is at our disposal. We have 

K n (a,z) \ 2 ds = K n (a,a), 



\ \y'(z)\ds= | 2 "d0=2,r, 

JC JO 



K n (a, z) {/ (z)}* ds = K n {a, g 



o 



I V f ^f 

I J^ < d 

where = e". Jo 

Now the function K n {a, g ()} vV (0 is regular and analytic for 
| | < 1, and so the last integral is equal to 



n {a, g (0)} V) = ^K n (a, a) [/ (a)]-. 
Choosing A - ~- [y (a)]*, 

we have finally J n = ^- | / (a) | ^T n (a, a). 

ZTT 

Our object now is to show that 

lim J n - 0. 

n->oo 

Let us write () = fo' ()]* 

Since gr' () ^ for | | < 1, a branch of L () is a regular analytic 
function for | f | < 1. 

We now consider the set of analytic functions E () regular in | | < 1 
and such that f2ir 

|L()tf(0|<M=l. 
Jo 



320 Conformal Representation 

Let a be a fixed number whose modulus is less than unity, then the 
maximum value of | E (a) \ 2 is 

[1- ||2]-i|L()|-2. 
To see this we put 

L(QE(l) = t + t l t+...+t n F+... 9 
then on the above supposition 



and Schwarz's inequality gives 

\E(a)\*\L(a)\*< \t n 



The sign of equality holds when, and only when, 
t n = ca", n=0,l, 2, .... 

that is, when L () # () = j-^^g. 

M f 2 ' ^ _ 271 L 

w Jo 1 1 ~si~ r-ii 2> 

therefore 27r | c | 2 = 1 - | a | 2 , 

and so JB ( 0-. (1 -* 



-.- -. 

1 - Jo (27T)*Zf(C) 

Now let (0 = -E {y (2)} = (? (z) = (? {g (J)}, 

then jE? (^) and (7 (2) are simultaneously regular, and 

1= f a '|L(0^(0| l de= [ 8 "|i7'(OII^U)|'de= f IGW*. 

Jo Jo Jc 

Finally, E (0) - (7 (a), 

so that max | E (0) | 2 - max | G (a) | 2 , 

therefore 

K (a, a) = max | G (a) |- = max | JB (0) | = ^-/^ = ^^ | , 

and so 

Km -/ = ^ | / () | - ^ (a, a) = ~ {| / (a) | - | g' (0) |} = 0. 

n -> oo ^^ ^ 7r 



Since [^ n (a, z) - A {y' (z)}Jp = F n (z) 

is a regular analytic function in the closed inner realm of (7, we have for 
any point z within C whose least distance from C is 8, 



and so as n ->oo, lim | .P n (z ) | = 0. 



Region Outside, a Closed Curve 321 



Hence K (a, z,) = lim K n (a, z ) = A {/ 

n->oo 

Furthermore, since 

K (a, a) - j^ | / ( 

we have / W - 2. I* 

o rg 

and so y (2) = ~ -r [K (a, z )] 2 dz Q . 

A (a, a) J a 

If the curve (7 instead of being of unit length is of length I the ortho- 
gonal polynomials P n (z) are defined so that 



and the general formula for the mapping function becomes 



where e is a number with unit modulus and is equal to unity when the 
mapping function is required to be such that y' (a) > 0. 

A study of the expansion of functions in series of the orthogonal poly- 
nomials P n (z) has been made recently by Szego and by V. Smirnoff, 
Comptes Rendus, t. CLXXXVI, p. 21 (1928). 

4-82. The mapping of the region outside C' . If we write 
z' (z a) = 1, ww' = 1, 

the interior of C maps into the region outside a closed curve C' in such 
a way that the point z = a maps into the point at infinity in the z'-plane. 
The interior of the unit circle | w \ < 1 is likewise mapped into the region 
| w' | > 1, the point w = corresponding to w' = oo. 

Hence the region | w r \ > 1 is mapped on the region outside C' in such 
a way that w' = oo corresponds to z' = oo, the relation between the 
variables being of type 

z' = rw' + T -f T! (w')~ l + ... + r n (w/)~ n -f .... 
Since w = y (z), the function which gives the conf ormal representation is 

w' = [y {a 4- (z')- 1 }]- 1 = (A say. 

Szego has shown that the function (z') may also be obtained directly 
with the aid of an orthogonal system of polynomials Yl n (z') associated 
with the curve C". 

If T = | T | 6 fa , we have in fact the formula 



322 Conformal Representation 

EXAMPLES 

1. If z is a root of the equation P n (z) = 0, prove that 



^--- I zas. 

Z -Z | 

Hence show that z lies within the smallest convex closed realm R which contains the 
curve C. 

2. If C is a circle of unit radius, 

P (z} z n 

L n \*f "~ * > 



3. If the curve C is a double line joining the points 1,1, the polynomial P (z) becomes 
proportional to the Legendre polynomial. Note that in this case the series K (a, z) fails to 
converge, but this does not contradict the general convergence theorem because now the 
points a and z do not lie within C. 

4. If jR n (z) is any polynomial and a any point within the curve (7, prove that 



4-91. Approximation to the mapping function by means of polynomials 
Let a circle of radius R be drawn round the origin in the z-plane and let 

w = / ( z ) = a o + a i z + a 2 z * + 

be a power series converging uniformly in its whole interior. This maps 
the circle on a region of the complex w-plane. For the area of this region 
we easily find the expressions 

A = I" [ 2?r I /' (z) \*rdrde (z - re) 

Jo Jo 



= 277 \ R 

Jo 



rdr S | a n 

n=l 

a l I 2 + TT S n\ a n | 2 R 2n + ... . 

n-2 

The area of the image region is always greater than 77.R 2 | a x | 2 when 
a x 7^= and is always greater than TTH | a n | 2 J? 2n when a n ^ 0. When the 
mapping function/ (z) is such that a : = 1 the result is that the area of the 
picture is greater than that of the original region unless the picture happens 
to be a circle of radius R. 

Suppose now that we are given a simply connected smooth limited 
region B of the z-plane. Let dr be an element of area of this region, we 
then look for a function / (z) regular in B which makes the integral 

7 =JJ |/'(S) |(*T 

* L. Bieberbach, Rend. Palermo, vol. xxxvm, p. 98 (1914). 



Approximation by Means of Polynomials 323 

as small as possible. To make the problem definite we add the restrictions 
that / (0) = 0, /' (0) = 1, and that / (z) is a polynomial of the nth degree. 
These conditions are satisfied by writing 

f(z) = z + a 2 z*+ ...+ a n z n . ...... (A) 

If / (z) -f g (z) is a comparison function we have to formulate the 
conditions that the integral 



(0 = j] I /' 



+ *g' (z) \ 2 dr^ [/' (z) + eg' (z)] [T(z) + eY~&} dr 



may be a minimum for = 0. These conditions are 
~ 



= g ' {z) * r w dr = I v' M I 2 dr - 



The inequality is always satisfied, but the two equations are satisfied 
for all forms of the polynomial g (z) only when the coefficients a s satisfy 
certain linear equations. If 



where z is the conjugate of z, these equations are 
2z 0>1 4- 4z ltl a 2 + 6z 2jl a 3 4- ... 2nz n _ lt l a n = 0, 



i + (n.%) Zi, n ~ia* + (n.'3) z^^a^ + ... (n.n) z n _^ n ^a n - 0, 
and 

2z lf0 4- (2.2)z 1>1 a 2 -f (2.3)z 1 , 2 a 3 + ... (2.n)z ltW . 1 o B = 0, 

^n-i f o + (ra.2) z n _ 1>]L a 2 -f (w.3) 2; n _ 1>2 a 3 + ... (n.n) z n _ lin ^d n = 0. 
These linear equations are associated with the Hermitian form 

n 2 ln 2 (p+ 1) (5+ l)^a^, 

p=0 g0 

and possess a single set of solutions for which / is a minimum. By giving 
different values to n we obtain a sequence of polynomials which in many 
cases converges towards a limit function F (z). The question to be settled 
is whether this function F (z), among all mapping func- ^ -\ 
tions with the properties F (0) = 0, F' (0) = 1, gives the f \ 

smallest possible area to the picture into which B is I ^ \ 

mapped. The following simple example tells us that this 

is not always the case. Consider the region B which 

arises from a circle when the outer half of xme of its 

radii is added to the boundary (Fig. 26). There is no Fi &- 26 ' 

polynomial which maps this region B on a region of smaller area. For 




324 Conformal Representation 

by means of a polynomial the region B is mapped on another region 
which has the same area as the region on which the complete circle is 
mapped and, unless the polynomial is simply z, this region has an area 
which is greater than that of the circle. Hence in this case all minimal 
polynomials are equal to z and F (z) is also equal to z. 

Bieberbach has investigated the convergence of the sequence of 
polynomials to the desired mapping function for the type of region 
discovered by Carath^odory*. For such a region the boundary is contained 
in the boundary of another region which has rib point in common with the 
first. The interior of a polygon is a particular region of this type and so 
also is the interior of a Jordan curve. 

Bieberbach's method of approximation has been used recently in 
aerofoil theory for the mapping of a circle on a region which is nearly 
circular f. 

Introducing polar co-ordinates, z = re ie , and supposing that on the 
boundary r = 1 -f y, where y is small, we may write 

T2TT fl+y I r2ir 

m }Q Jo 2> 4- <7 + 2 J o 

Hence retaining only terms up to the second order in the binomial 
expansion of (1 + y)p+<J+ 2 ? 

Z PQ= \ y-cos (p - q) d.dO + P -f y 2 .cos (p - q) 0.d0 

Jo * J o 



(p - q) d.dd + -- 2 .sin (p - q) 0.d0, p 



These quantities may be determined from the profile of the nearly 
circular curve when this is given. 

Now writing z w - f w + irj^, a p = <f> p + i$ p , 

where f OT , 77^, <f> p and ifj p are all real, and neglecting all the coefficients 
after a 4 in the expansion (A), we obtain the following equations for the 
determination of </> 2 , </> 3 , </> 4 , i/j 2 , 3 , i/r 4 : 

01 + 2 ii</2 + 3 12 (/> 3 -f 3^^ -h 4| 13 </> 4 -f 47? 13 </r 4 = 0, 

^w + 2f u ^ a - 3^ 12 ^ 3 -f 3| 12 </r 3 - 47 ?13 ^ 4 + 4^ 13 4 = 0, 

^02 + 2^ 12 ^ 2 ~ 2^^ + 3f 22S 6 3 -f 4 23 < 4 + 4^23^ - 0, 

??02 + 2^12^2 + 2 ^12^2 + 3f 22 </T 3 ~ 47723(^4 + 4^23^ - 0, 



%j + 277 13 ^ 2 + 2 13 </r 2 -f 37723^63 + 3^^ -f 4fa 

* Jtfo^. ^Inn. vol. LXXII, p. 107 (1912). 

t F. Hohndorf, Zeite. f. any. Math. u. Mech. Bd. vi, S. 265 (1926). The conf ormal representation 
of a region which is nearly circular is discussed in a very general way by L. Bieberbach, Sitzungsber. 
der preussischen Akademie der Wissenschajten, S. 181 (1924). 



DanielVs Orthogonal Potentials 325 

Eliminating 4 and </r 4 we obtain the equations 
(0133) + 2 (1133) fa +3 (1233) fa 4 3 [1233] </ 3 = 0, 

[0133] + 2 (1133) fa - 3 [1233] < 3 + 3 (1233) ^ 3 - 0, 

(0233) + 2 (1233) fa - 2 [1233] 2 + 3 (2233) < 3 - 0, 

[0233] + 2 [1233] </> 2 4 2 (1233) 2 4-3 (2233) </r 3 - 0, 

where 

(jpgr*) - $$ - ^ f w - rj pr Tj qs , [pqrs] = 17^17,., 4 f^ifc, - i? pr f w , 
and these finally give the values 

vNfa = Z 2 ,_ 3 , yAty, - Z 2 ,_, , v = 2, 3, 4, 
where N = (1233) 2 4- [1233] 2 - (1133) (2233), 

Z l = (0133) (2233) - (0233) (1233) - [0233] [1233], 
Z 2 - [0133] (2233) 4 (0233) [1233] - [0233] (1233), 
Z B - [0133] [1233] 4 (0233) (1133) - (0133) (1233), 
Z 4 - [0233] (1133) - (0133) [1233] - [0133] (1233). 

4- 92. DanielVs orttiogonal potentials. Consider a set of polynomials 
p Q (z), Pt (z), ... defined by the equations* 

(0,0) (1,0) (71,0) 

(0, 1) (1, 1) (n, 1) 



where 



(0, 71-1) (1,73 
1 

(0,0) (0,1) (0,71) 
(1,0) (1,1) (l,w) 



(71,71- 

z n 



and 



(n,0) (TI,!) (7i, n) 
(m, n) = ^- z m z n dr, 

the integral being taken over the region to be mapped on a unit circle. 
A denotes here the area of the region and dr an element of area enclosing 
the point z. These polynomials satisfy the orthogonal conditions 

\ r f _ 

3 Jj Prn (z) Pn (*) dr - 0, m^n 

= 1, m = n. 

A mapping function / (z) which satisfies the conditions / (a) = 0, 
/' (a) = 1, is given formally! by the expansion 



4 



W f 

J a 



Pl (z') 



where 



fif = p (a) p (a) + 



(a) 4 .... 



* These equations are analogous to those used by Szego. 

f This series does not always represent an appropriate mapping function as may be seen from 
a consideration of the circular region with a cut extending half-way along a radius as in 4-91. 



326 Conformal Representation 

To see this we write 

/' (z) = flo^o (z) + aiPi (z) 4- a 2 p 2 (z) 4- ... , 



(z) 

where a , a x , . . . ; a , a x , ... are coefficients to be determined, so that the 
integral 



f (z)f'(z).dr = a a + a,a t -f ... 
may be a minimum subject to the conditions 



/'(a)=l, />)=!. ...... (A) 

Differentiating with respect to a , a l9 ... ; a , a l9 ... in turn we find that 

n = &Pn (), W = 0, 1, 



where i and ^ are Lagrangian multipliers to be determined by means of 
the equations (A). We easily find that 

1 = kS, kS = 1, 
and so 

Sf (z) = Po (a) Po (z) + ^1S) Pl (2;) + ... . 

If p n (z) = u n ~ iv n , where u n and v n are real potentials which can be 
derived from a potential function <f> n by means of the equations 

_ d<f, n ty, 

M " ~ dx ' " ~ dy ' 

we have U[ (-^ ^ + ^ ^ dr = 0, m^ 

J[ J J \ 3x 3o; dy dy / 

= 1, m = 7i. 

The potentials </> , </> 1? ... thus form an orthogonal system of the type 
considered by P. J. Daniell*. This definition of orthogonal potentials is 
easily extended by using a type of integral suggested by the appropriate 
problem in the Calculus of Variations. 

For the unit circle itself the orthogonal polynomials are 

Pn(*) = &.(n+ 1)*, 
and the mapping function is consequently given by the equations 



f( z \ = v 1 - au ) v z - <*) 
4-93. Fejer's theorem. Let 

be the function mapping a region D in the Z-plane on the unit circle d with 

* Phil. Mag. (7), vol. ii, p. 247 (1926). 



Fejer's Theorem 327 

equation | z \ < 1 in the z-plane. We shall suppose that D is bounded by a 
Jordan curve C and that Z = H (9) is the point on G which corresponds to 
the point z = e ie on the unit circle c which bounds the region d. At this 
point, if the series converges 

Z = a Q + a^ 19 -f a 2 e 2ie + ... a n e in * + ... 

= ^o + Wi + ^2 + say. ...... (2) 

Now by Cauchy's form of Taylor's theorem 



= 0, n < 0, 

where n is an integer and the contour is a simple one enclosing the origin 
and lying within the circle of convergence of the power series. On account 
of the continuity of / (z) in d we may deform the contour until it becomes 
the same as c without altering the values of the integrals. Hence, writing 

= e m we get 

f2jr foo 

2?7 a n = H (a) e~ ina n > 0, = // (a) e ina da. 
Jo Jo 

These equations show that the series (2) is the Fourier series of the 
continuous function H (6) and is consequently summable (<7, 1) ( 1*16). 

Now consider a circle j z \ = p where p< 1. The function / (z) maps the 
interior of this circle on the interior of a region R whose area A is, by 
4-91, equal to the convergent series 

7 r[|a 1 |V+2|a 2 |V+-"]- 
This area A is bounded for all values of p and is less than B, say, 

' "[KIV + 2 I 2lV + ...n\a n \*p* n ]<B 
for < p < 1 and so 

7r[|a 1 | a +2|o 2 | 2 + ...n\a n \*]<B. 

This inequality shows that the series S n \ a n | 2 is convergent. Now this 
property combined with the fact thfrt (2) is summable ((7, 1) is sufficient 
to show that the series (2) converges. Writing 

s n - U Q + u^ + ... u n , (n + 1) S n = 5 + $! + ... s n 

we have S n = s n - a n where (n + 1) cr n = t^ + 2^ + ... nu n . It is suffi- 
cient then to show that a n -> as /i -> oo. With the notation # n = | i^ n | 
we have the inequality 

[(m f- 1) v m+1 -f ... (m + p) v m + P ] 2 

< [(m + 1) + ... (m + p)] [(m + 1) v 2 m+1 + ... (m + p) v* m+J> ]. 

The first factor on the right is less than 1 -f 2 4- ... (m + p) which is less 
than (m + p + I) 2 . Also, since the series 2rw n 2 converges we can choose 



328 Conformal Representation 

m so large that the second factor is less than e 2 whatever p may be. We 
may now write n = m + p, 

* = v\ + 20 2 + mv m (w + 1) v m+l + ... (m + p) v m+J> 
n ~~ m -}_ p -j- i m _+_ p -f 1 > 

where the second term on the right is less than e and since p is at our 
disposal we may choose it so large that the first term on the right is less 
than e. Hence we can choose n so large that | v n \ < a n * < 2e and so 
| a n | -> as n -> oo. 

It follows then that the series (2) converges and that the co-ordinates 
(X, Y) of a point on a simple closed Jordan curve can be expressed as 
Fourier series with 6 as parameter. The theorem implies that the series (1) 
converges uniformly throughout d and that the mapping by means of the 
function / (z) may be extended to regions which are slightly larger than 
d and D. 

Reference is made to Fejer's papers, Milnchener Sitzungsber. (1910), 
Comptes Rendus, t. CLVI, p. 46 (1913) for further developments. Also to the 
book by P. Montel and J. Barbotte, Lemons sur les families normales de 
fonctions analytiques (Gauthier-Villars, Paris, 1927), p. 118. 



CHAPTER V 
EQUATIONS IN THREE VARIABLES 

5-11. Simple solutions and their generalisation. Commencing as before 
with some applications of the simple solutions we consider the equation 



_ 

p W ~ ^ \d& + 

of the propagation of Love-waves in the direction of the #-axis. If now 
p and fjb have the values p , /ZQ respectively for z > 0, and the values p , p^ 
respectively for z < 0, it IP useful to consider a solution of type 
v = v = (A cos sz -f B sin sz) sin K (x qt) 3 - z > 0, 

v = Vl = C f c to sin K (x gtf), 2 < 0, 
where the constants are connected by the relations 

A*V = Mo (* 2 + * 2 )> Pi^V = Mi (^ 2 ~ * a )- 

In the expression for v 2 we take h > so that there is no deep pene- 
tration of the waves. 

The boundary conditions are 

/^ -~ = 0, when z = a, 




These equations give 

4 sin sa = J5 cos sa, 

4 = (7, 



Putting ^ = C 2 p , ^ = q^! we have with s = K cosech o> , A = K sech o^ , 
c = g tanh o; , c x = y coth co x , 

cosh Wi = sinh o> cot (CLK cosech co ) = ( 1 ^ tanh 2 o> ) , 
Mo V c i ' 

and it is readily seen that there are no waves of the present type unless 
c <c 1 . 

Matuzawa has examined the case of three media arranged so that in 
his notation 

v = v l = A l (ei z + e-*i z ) cos (pt + fx), p = p 1? M = Mi> > 2 > - A, 
v = v z = (^2 e ** z + Be~'* z ) cos (jp -f /), p = />2J M == M2> h> z> H, 

V = V 3 = ^36*8* COS (p^ -h /#), P = P3> M^MSJ "" H > Z. 



330 Equations in Three Variables 

The boundary conditions 

dv l dv z . , 

^i - v 2 , to - = ft ~ at z - - 



give ^4j (e~*i A + 



Eliminating ^ij, ^4 2 ^3> ^2 an d writing r x = tanh sji, r 2 = tanh s 2 h, 
T 2 = tanh s 2 H, we obtain the equation 



? (T 2 - T,) 



The cases ^, 5 2 , 3 all real and 5 2 imaginary, s l9 s 3 real, are not com- 
patible with an equation of this type. When s 2 is real it appears that there 
is only one value of s l and this is an imaginary quantity; when .$ 2 is an 
imaginary quantity it appears that there are two possible values of s l and 
these are both imaginary. 

Matuzawa has examined the six possible cases 
A B G D E F 

C l <C 2 < C 3 C l <C 3 < C 2 C 2 <C l < C 3 C 3 < G! < C 2 C 2 <C 3 < C t C 3 < C 2 < C x 

and concludes that in cases B, D and F there is no solution. 

5*12. The simple solutions considered so far correspond to the case 
of travelling waves. We shall next consider a case of standing waves and 
shall take the equation of a vibrating membrane 



dt 2 ~~ \dx* 

Let the boundary of the membrane consist of the axes of co-ordinates 
and the lines x = a, y = b. The expression 

. m-rrX . UTTV , 4 ^ . .. 

w = sin sin -^ {^4 mn cos pt + B mn sin jrf} 
w c/ 

satisfies the condition w = on the boundary and is a simple solution of 
the differential equation if 



This equation gives the possible frequencies of vibration, ra and n 
being integers. A more general type of vibration may be obtained by 
summing with respect to m and n from m = 1 to oo and n = 1 to oo. 



Standing Waves 331 

The resulting double Fourier series is usually a solution which is sufficiently 
general to make it possible to satisfy assigned initial conditions 

w ^ WQ ' dt = ^ f r ^ ==0> 

by using coefficients A mn , B mn determined by Fourier's rule 
4 4 [ a f & . mrrx . 



a 6 nny 

- 



4 f a f 6 . mTTX . 
n = ~T w o sin s 

a6jpJoJo a 



EXAMPLE 

Find the nodal lines of the solutions 



. . TTX . *rrtj f nX 7rtJ\ 

> = A sm - - sm -- (cos h cos ) cos pt, 

a a \ a a / ^ 



A . '27TX . 2rry 

w = A sin sin - cos pt, 
a a 

~ ( . SKX .Try . TTX . STT^I 
w C {sin sin - sin sin -> coapt, 

\ a a a a j 

which are suitable for the representation of the vibration of a square if p has an appropriate 
value in each case. 

5-13. Reflection and refraction of electromagnetic waves. In a non- 
conducting medium the equations of the electromagnetic field are* 
curl H - J9/c, div D = 0, 
curl E = - B/c, div B = 0, 
and the constitutive relations are 

D - xE, B= /*//, 

where the coefficients K and ^ can be regarded as constants if the material 
is homogeneous and the frequency of the waves is not too high. 

If all the field vectors are independent of z, their components satisfy 
the two-dimensional wave-equation 



where V 2 = c 2 /Kp, = 1/s 2 , say. 

The permeability of all substances is practically unity for frequencies as 
great as that of light. Hence for light waves it is permissible to write 

V = C/VK, 

and in this case we may also write /c = n 2 , where n is the index of refraction 
of the medium. 

Let us now suppose that the medium with the constants (K^, /^) is on 

* For convenience we denote a partial differentiation with respect to the time t by a dot. 



332 Equations in Three Variables 

the side x < of the plane x = 0, and that on the other side of this plane 
there is a medium with constants (/c 2 , /x 2 ). 

We shall suppose that when x < there is an incident and a reflected 
wave, but that for x > there is only a transmitted wave. We shall 
suppose further that the electric vector in all the waves is parallel to the 
axis of 2, then with a view of being able to satisfy the boundary conditions 
we assume 

E z - A& + AM (x < 0), E, = A 2 e 2 (x > 0), 

where e^, e ', e 2 denote respectively the exponentials 

g __ gitofs^j; co8#i + 1/ sin <>!)-'] g ' 



The corresponding expressions for the components of H are 
H x - (csjfr) (A& 4- AM) sin </>!> # < 0, 
H x = (c5 2 //LL 2 ) ^4 2 6 2 s i n </>2> a; > 0, 

// = (c$i/ /L4) (A'e/ ~ ^i e i) cos </>!, a < 0, 
H y = (cs 2 /iJL 2 ) A 2 e 2 cos </> 2 , a; > 0. 

The boundary conditions are that the tangential components of E and 
// are to be continuous. These conditions give 

A.} -f- AI = A 2 , 
sin fa.ptSi (A l -f AI) = /^i 2 ^2 sin ^2 

COS <>i.X 2 <Sl (-^1 ^/) = ^1 5 2^2 COS <2- 



TT l l2 2 l 

Hence . ^ = ^- = when u x = u 2 . 

sin (f> 2 p, 2 s i n i 



This is the familiar relation of Snell. Writing 

A 9 = fi-4, ^ 2 = T^, 

where 7? and T are the coefficients of reflection and transmission re- 
spectively, we have 

1 R _ sin ^ cos fa T* _ g i n (^i ^2) 

1 -}- jR sin <^ 2 cos </>! ' sin (0 t + ^2) ' 

2 T sin <^ x cos ^> 2 ^ 2 sin ^> 2 cos ^ 
2 7 ~" sin ^ 2 cos </>! ' ~~ sin (fa 4- <^ 2 ) 

In the case when the electric vector is in the plane of incidence we write 
E X - - (C& + CM) sin <i> EX = - ^2^2 sin </> 2 , 

^ = ((7^ - CW) cos ^ (a; < 0), E v --= ^63 cos < 2 (x > 0), 
H, = (c^/^) (O lCl + CM), H z 

and the boundary conditions give 



GI) cos ^ = (7 2 cos ^ 2 
-f C f 1 / ) sin </>! = ^^ sin 



Reflection and Refraction 333 

The third equation implies that the ^-component of D is continuous 
at x = 0. These equations give 
sin <A 1 



= 
Sin 



Thus Snell's law holds as before. If we further write 

CS-pCv C t =rC t , 
so that p, T are the coefficients of reflection and transmission respectively, 

WC haVe = tan (A - c 2 ) 

p tan (^ + </> 2 ) ' 

2 sin < 2 cos (/>! 
r ~ sin (<j, + <f> 2 ) cos (</>! - ^> 2 ) ' 

Of the four quantities R, T, p y r only one can vanish, viz. the polarizing 
angle O x is defined as the angle of incidence for which p = 0. This angle 
is given by the equation tan (^ -f- <f> 2 ) = oo, and so 

tan Oj n 2 /n 1 . 

When the incident light is unpolarized it consists of a mixture of waves 
in some of which E is parallel to the axis of z and in the others H is parallel 
to the axis of z. When such light strikes the surface x =0 at the polarizing 
angle the waves of the second kind are transmitted in toto, and so the 
reflected light consists merely of waves of the first kind and is thus linearly 
polarized. 

Reflection and refraction of plane waves of sound. Consider a homo- 
geneous medium whose natural density is p . When waves of sound 
traverse the medium the density p and pressure p at an arbitrary point 
Q (%> y, z) have at time t new values which may be expressed in the forms 

p = Po(l + 5), p = p (l 4- As), 

where p is the undisturbed pressure and A is a coefficient depending on 
the compressibility. The quantity s is called the condensation and will be 
assumed to be so small that its square may be neglected. 

We now suppose that the velocity components (u, v, w) of the medium 
at the point Q can be derived from a velocity potential <f> which depends 
on the time. Bernoulli's integral 

[dp d<f> , . 

\ - + -7T7 = constant 
] P dt 

then gives the approximate equation 



where c 2 = p Q A/p Q = (dpjdp) Q is the local velocity of sound and is constant 
since A and p Q are constants. The equation of continuity 



334 Equations in Three Variables 

and the equations u ~ ~ , v = -~ , w = ^, when ?/, t;, w are small, give 



the wave-equation 



<>i 



The conditions to be satisfied at the surface separating two media are 
that the pressure and the normal component of velocity must be con- 

tinuous. On account of Bernoulli's equation the continuity of pressure 

p / 

implies that p ^~ is continuous. 

Let us now consider the case in which two media are separated by the 
plane x 0. We shall suppose that in the medium on the left there is an 
initial train of plane waves represented by the velocity potential 

^ = aoC-f->, 

and that these waves are partly reflected and partly transmitted. We 
therefore assume that 

(/>! = a e tn(t -t x - } + a 1 e in(e+ffl? - 1iy) , x < 0, 
</> 2 = a 2 e tn(t -S x - } , x > 0. 

The boundary conditions give 

Pi (0'0+ %) 



where p l and p 2 are the values of the natural density for x < and x > 
respectively. If c x and c 2 are the two associated velocities of sound, we 
must have c^cos^, c^sin^, 

c 2 = cos 2 > C2 7 ? ^ s i n a 2 
Therefore 



__ cos ! c l p l gos 2 ^2 c t a i ~" Pi 

1 COS a l + C lPl COS ^2 P2 CO ^ a l + Pl 

cos % _ 2/>! cot 

" - " ~ " 



ri . . 

COS ! -h C 1 p l COS a 2 /0 2 COt ! -f- /)j COt a 2 

The equation c 2 sin % = c t sin 2 

gives a law of refraction analogous to Snell's law. 

When the second medium ends at x = 6, where b > 0, and for x > b 
the medium is the same as the first, there are three forms for the velocity 
potential: ^ ^ ^ginu-f*-) + a ^ in (t^ x -^ y ^ x < 0, 

</> 2 = a 2 e in(t "^-^ + a 3 e inU +^-^>, 6 > a; > 0, 
(/> 3 - a 4 e mU -^-^>, 
and the boundary conditions give 

pi K -f i) == p 2 K + 3) ^ K - i) = (2 - %)> 



Energy Equation 335 

Therefore 

cos (*6) + -^ sin 

4 

; s in (n&) 4- ^ 
4 

= os (n) + Jt i 2 + sn 
- 2 - M sin 

bPl P2J 

It should be noticed that these equations give 

Kl 2 - I oil 2 - Kl 2 , 

I 19 I 19 I 19 

p 2 1 2 1 2 - /> 2 1 % r = PI I ^ r cot a , 

and the first of these equations indicates that the sum of the energies per 
wave-length of the reflected and transmitted waves in the first medium is 
equal to the energy per wave-length in the incident wave. 

It should be noticed that if sin (nb) ^ 0, the condition for no reflected 
wave (% = 0) is p 2 ~ pi> and is independent of the thickness of the 
second medium. 

We have assumed so far that there is a real angle 2 which satisfies the 
equation c sin 0% c 2 sin x , but if c 2 > Cj it may happen that there is no 
such angle. If the value of sin 2 given by this equation is greater than 
unity, cos 2 will be imaginary and the solution appropriate for a single 
surface of separation (x 0) will be of type 

</> x - a e in(t ^ x -^ -f a 1 e tnU+ ^- Tjy) , x < 0, 

^ 2 = o 2 e t " n( *- 1 ' ) -* a! , x > 0, > 0. 

In this case there is no proper wave in the second medium, and on 
account of the exponential factor e~ 6x the intensity of the disturbance falls 
off very rapidly as x increases. The corresponding solution of the problem 
for the case in which the second medium is of thickness b is obtained from 
the formulae already given by replacing by id. It is thus found that 



P* 



-- 
LP2 0J 

a 4 [cosh (nbO) 4- \i (^ - ^} sinh (nb0) I , 

L Wl Sp2/ J 

2 -f ^l sinh (nbO). 



The coefficient a 3 of the disturbance of type e inu " liv)+a * which increases 
in intensity with a; is seen to be very small so that this disturbance is small 
even when x = 6. 



336 Equations in Three Variables 

In the present case the reflection is not quite total, for some sound 
reaches the medium x > b. The change of phase on reflection is easily 
calculated by expressing aj/a in the form jRe*". 

Let us now consider briefly the case when nb krr, where k is an 
integer. In this case sin (nb) = 0; there b no reflected wave and the 
formulae become simply 

</i = a^e in(t ~^-^\ x < 0, 

</> 2 = o Wp2) e in(t -* x -> + a 3 e in(t ~> sin (nx), b > x > 0, 
(f> 3 = a e tnU ~ f *-' I1 ' ) , x > b. 

It will be noticed that the value of n is precisely one for which there is 

a potential </> fulfilling the conditions -~ = for x = and x = b. 

The slab of material between # = and x = b can be regarded as in a 
state of free vibration of such an intensity that there is no interference 
with the travelling waves. 

The absorption of plane waves of sound by a slab of soft material has 
been treated by Rayleigh* by an ingenious approximate method in which 
the material is regarded as perforated by a large number of cylindrical 
holes with axes parallel to the axis of x and the velocity potential within 
these holes is supposed to satisfy an equation of type 



where h is a positive constant. The new term is supposed to take into 
consideration the effect of dissipation. 

At a very short distance from the mouth (x = 0) of a channel it is 

}2JL PV2JL 

assumed that the terms ~ - 2 and ^ ^ may be neglected and that the 
solution is effectively of type 

= e int {a! cos k'x -4- 6' sin k'x}, 
where c 2 k' 2 = n 2 inh. 

If the channel is closed at x = 6, we have ^ - = there, and so we may 

write 

</ = ^4V n 'cosfc' (x b). 

When x is very small 



C 2 5 = _ _r = _ inA'e ini cos (k'b), 
u ik' 



c*s n {k ' b} - 

* Phil. Mag. (6), vol. xxxix, p. 225 (1920); Papers, vol. vi, p. 662. 



Reflection 337 

If, for x < 0, we adopt the same expression as before, viz. 



we have 

Now let a be the perforated area of the slab and v' the area free from 
holes. The transition from one state of motion on the side x < to the 
other state on the side x > is assumed to be of such a nature that 

(cr -f a') u l = cm, 



These equations give the relation 
a a* ik f 



.-. ,, . 
, tan (kb) 



a a i n $ <* 

for the determination of the intensity of the reflected wave. When h 0, 
we have | a x | = | a | and the reflection is total, as it should be. When 
a = 0, a = a , and there is again total reflection. On the other hand, if 
a = 0, the partitions between the channels being infinitely thin, we have, 
when h = 0, * 

_ ng cos (k'b) ik' sin (k'b) __ cos j cos (k'b) i sin (k'b) 
1 ~ ?i cos (&'6) -f ^' sin (k'b) ~ cos j cos^F6) 4- * sin (k'b) ' 



In the case of normal incidence x = 0, a x = a e~ 2ifc/b , and the effect is 
the same as if the wall were transferred to x = b. When h is very small 
but the term k 2 in the complex expression k' = k^ + ik 2 is so large that 
the vibrations in the channels are sensibly extinguished before the stopped 
end is reached, we may write 

cos (ik 2 b) = Je fc 2 & , sin (ik 2 b) \ie k ^, tan (k'b) = i, 
and the formula becomes 



a + di (cr + a') cos a ' 

EXAMPLES 

1. In the reflection of plane waves of sound at a plane interface between two media 
the velocity of the trace of a wave-front on the plane interface is the same in the two media. 

* [Rayleigh.] 

2. When the velocity of sound at altitude z is c and the wind velocity has components 
(u, v, 0), the axis of z being vertical, the laws of refraction are expressed by the equations 

<f> = <t> Q , c cosec 9 -h u cos <f> + v sin <f> =-- c cosec + U Q cos < + v sin < = A, say, 

where (0, <f>) are the spherical polar co-ordinates of the wave-normal relative to the vertical 
polar axis and the suffix is used to indicate values of quantities at the level of the ground. 

3. Prove that the ray- velocity (the rays being defined as the bicharacteristics as in 
1*93) is obtained by compounding the wind velocity with a velocitj 7 c directed along the 
wave-normal. See also Ex. 1, 12-1. 



338 Equations in Three Variables 

4. The range and time of passage of sound which travels up into the air and down again 
are given by the equations 

rz 

x = 2 (c 2 cos < -f us) dz/T, 
J o 



fZ 

y = 21 (c 2 sin <f> + vs) dz/r, 
J o 



fZ 

* = 2 sdz/r, 
Jo 

where s A u cos <f> ~ v sin <f>, r s (s 2 c 2 )*, 

and Z is defined by the equation s = c. 

5-21. Some problems in the conduction of heat. Our first problem is to 
find a solution of the equation 



which will satisfy the conditions 

6 exp [ip (t x/c)] when y = 0, 6=0 when y = oo. 
Assuming as a trial solution 

exp [ip (t - &/c - y/b) - ay], 

Kip\ 2 7? 2 ~l 
a + ~-j ^-g 

Therefore 6 = 2a/c, p 2 ( ~ -f ~) = a 2 . 

-r ^ 2 c ay 

The result tells us that if the temperature at the ground (y = 0) varies 
in a manner corresponding to a travelling periodic disturbance, the variation 
of temperature at depth y will also correspond to a periodic disturbance 
travelling with the same velocity but this disturbance lags behind the 
other in phase and has a smaller amplitude. 

The solution may be generalised by writing 

b = c tan </>, a = (c/2/c) tan <, p = (c 2 /2/c) tan (/> sin 0, 

7T 

f2 

# = f (<f>) d<f>. exp [(ic/2/c) (c x) tan <f> sin <^ (cy/2/c) (tan (/> + i sin ^)], 
Jo 

where c i^> regarded as a constant independent of (f> and / (c/>) is a suitable 
arbitrary function. 

If we wish this solution to satisfy the conditions 

= g (ct x) when y = 0, = when y = oo, 
the function / (<) must be derived from the integral equation 



f 

= 

Jo 



2*) tan <f> sin </>], ( oo < u < oo). 



Solution 'by Definite Integrals 339 

When the function g (u) is of a suitable type, Fourier's inversion 
formula gives 

foo 

/ (<) ^ (c/47T/c) (1 4- sec 2 </>) sin <f>.g (u) du.exp [ (icu/2f<) tan </> sin </>], 

J-oo 

< </> < 7T/2. 

In particular, if 

</ (tf) = (2K/cu) sin [tan a sin a (c^/2/c)], 
where a is a constant, we have 

6 = f "sin <f> (1 + sec 2 0) dJ> . e~ <<*/2>tan* 
Jo 

x sin [(c/2/c) {(cZ x) tan </> sin </> y sin </>}]. 

Another solution may be obtained by making c a function of <f> and then 
integrating ; for instance, if c = 2/c cos (/> we obtain the solution 

TT 

f2 

= f ((f>) d<f>. exp [i sin 2 <f> (2i<t cos (^ x) y (sin ^ -f i sin <f> cos <^)]. 
Jo 

It should be noticed that the definite integral 

7T 

f2 

(#, y,z,t)= f((f))d</). exp [i sin 2 (2/c^ cos 6 x) z sin </> iy sin <b cos <i] 

Jo 

is a solution of the two partial differential equations 



and is of such a nature that the function 

9 (x, y, t) = (x, y, y, *) 

is a solution of equation (A). It is easy to verify, in fact, that if (x, y, z, t) 
is any solution of equations (B) the associated function (x, y, I) is a 
solution of (A), for we have 

d*e a^^a 2 a 2 a 2 a 2 

3z 2 + a*/ 2 ~ a* 2 + dy* + dz 2 dydz 

= 2 = 1 = 1 8 ^ 
dydz K dt K dt ' 

Again, if we take c = 2/c cot <f>, we obtain an integral 

IT 

(2 

(x, y, t) = / (</>) d<f>.exp [i (2.Kt cos (f> x sin </>) y (I -f i cos <^>)], 
Jo 

which is a solution of (A), and the associated integral 

7T 

(2 

(x, y, z,t)=\f (<f*) d<f> . exp [i (2/c cos <j> x sin <f> y cos <) ^- 2] 
Jo 

...... (C) 



340 Equations in Three Variables 

is likewise a solution of the equations (B). Indeed, if c is any suitable 
function of cf> the integral 



r 

= 
J 



(ct #) tan </> sin </> 
(c/2/c) (2 tan (f> + iy sin </>)] 



is a solution of the equations (B). 

It should be noticed that the particular solution (C) is of type 

(x, y, z, t) = e~*F (x, y - 2 K t), 
where F (u, v) is a solution of the equation 



This indicates that if F is any solution of this equation, then the 
function 9 ^ y ^ t) = \_ vp {Xt y _ 2Kt) 

is a solution of the equation (A). This is easily verified by differentiation. 
Since there is also a solution = er Kt F (x,y) 9 we have two different w r ays 
of deriving a particular solution of the equation (A) from a particular 
solution of the equation (D). 

Since F (u, v) = J \/\u* -f v 2 ] is a particular solution of equation (D) 
there is a certain surface distribution of temperature 

6 ^ J V|z 2 -f 4fc 2 * 2 ], when y = 0, 

which is propagated downwards as a travelling disturbance gradually 
damped on the way, the velocity of propagation being 2/c. 

If, on the other hand, we take F (u, v) = cos mu.exp v [m 2 l]i, we 
obtain a distribution of temperature 

(x, y, t) = e- cos mx.exp {(y - 2*0 [m 2 - 1]!}, m 2 > 1 ...... (E) 

in which a periodic surface distribution is decaying at the same proportional 
rate at every point of the surface. If m 2 < 2 the foregoing distribution 
gives = when y = oo. The periodic distribution now travels upwards 
with constant velocity 

c - 2* (m 2 - !)*/[! - (m 2 - 1)*], 

and the rate of damping at depth y is the same as that at the surface, but 
at any instant the temperature at this depth is a fraction 

exp[- 1 + (m 2 - 1)4] 
of that at the surface. When m 2 = 2 there is a distribution of temperature 

9 - e- 2 *< cos (x -v/2), 

which is independent of the depth but does not satisfy the condition* 
0=0 when y = oo. When m 2 > 2 the distribution (E) gives 6=0 when 

* In this case there is no solution of type = e~ 2ltt Y (y) cos (x N ; 2) which gives the foregoing 
surface value of t and a value 0=0 when y = oo for Y" (y) 0. 



Circular Source of Heat 341 

y = oo, and the material into which conduction takes place may be 
supposed to be on the side y < 0. In this case the velocity of propagation is 

c = 2* (m 2 - l)4/[(ra 2 - 1)* - 1], 
and the temperature at depth | y \ is at any instant a fraction 

exp - [(m 2 - 1)1 - 1] 
of that at the surface. 

We have seen in 2*432 that if (x, y, t) is a solution of equation (A) 
then the function 



9 -, 

is a second solution. If, in particular, we take the function 

6 (x, y, t) = er F (x, y), 
where F (u, v) satisfies (D), we obtain the solution 



If r 2 = x 2 H- i/ 2 there is a solution 



</> = ^e 4 ^ J (arlt) ...... (G) 

depending only on r and t which at time t is zero at all points outside 
the circle r = 2a/c. When t > the temperature at points of the circle is 
given by </> = t~ l J (2a 2 K/t). The circle can thus be regarded as a source of 
fluctuations in temperature which are transmitted by conduction to the 
external space. The total flow of heat from this circular source in the 
interval t = to t = oo may be obtained by calculating the integral 



dt. 



Now = - J ( 



Also [^ dt (alt 2 } e/ ' (2a 2 K/t) = - l/2*a, 

Jo 

r 

eft (a/J 2 ) J (2a 2 */) - 1/2/ca. 
Jo 

Hence dt ( ^ } = l//ca. 

Jo \^r) r ^ a< 

and so the total flow of heat from the circle is 



342 Equations in Three Variables 

This is independent of a and so our formula holds also for a point 
source. The temperature function of a point source of " strength" Q is thus 

^ = (ty**)- 1 ^/ 4 * ...... (H) 

while that of a circular source is 



)'i e J (ar/0- ...... (I) 

This result is easily extended to a space of n dimensions, thus in three- 
dimensional space the temperature function for a spherical source of 
strength Q is 



t)-*e * sm(ar/t)/(ar/t). ...... (J) 

The solution for an instantaneous source uniformly distributed over 
a circular cylinder has been obtained by Lord Rayleigh* by integrating 
the solution for an instantaneous line source. The result is 

r 2 + a 2 - 2ar cos r 2 -f a 8 



A more general solution is 

H + a 2 



Integration with respect to t from to oo gives a corresponding solution 
of Laplace's equation and we have the identity 



dt 

I I \ o -*-"" I I ty ^ ft \ 

* * I /-k . I C* . I I * ^-- t/, | 

(M) 

r > a. I 



The temperature due to an instantaneous line doubletf of strength q 
may be derived by differentiating with respect to y the temperature </> 
due to an instantaneous line source of strength q. Since the latter is 

<f> - (g/4^Kt) e-' 2 / 4 "', 
we have 6 = (qy^TTKH^) e~ r2 l M . ...... (N) 

The temperature due to a continuous line doublet of constant strength 
Q is obtained by integrating with respect to t between and t. Denoting 
this temperature by we have 

= 'flcft = (qylZtTKr*) ' ~ [e-*l*"] dt = 



= f'flcft = (qylZtTKr*) f ' ~ [e-*l*"] 
Jo Jo at 



* Phil. Mag. vol. xxn, p. 381 (1911); Papers, vol. vi, p. 51. 

t See Carslaw's Fourier Series and Integrals, p. 345 (1906). The direction of the doublet is 
that of the axis of y. The doublet is supposed to be "located" at the origin. 



Instantaneous Doublet 343 

This solution may be used to find a solution of (A) which takes the 
value F (x) when y = and is zero when y = oo and when t = 0. If 9 is 

to be such that 9, ~- and -5- are continuous for y > 0, an appropriate 
expression for 9 is 

00 rJ f (X-X* 

1 y a sec a a \ (0) 

& __ _ \ / 

da F(x-\- i/tana)e 4l<t . 



- 

Tt] 



2 

The first integral evidently satisfies (A) if y > 0, and the second integral 
tends to F (x) as y -> if F (x) is a continuous function of x. 

In the special case when F (x) = 1 the expression for 9 takes the form 

^y* 8ec a a 

""~ 



a 
and can be expressed in the well-known form* 

277-* e~ v2 dv, (P) 



where u 2 = y 2 /4:t<t and w > 0. 

If the boundary y = is maintained at the temperature F (x, t) the 
solution which is zero when y = oo and when Z = is given by the formula 



/ l'^' dt ' 

o (^ - t Y 

There is a similar formula for a space of three dimensions. 
If 9 = F (x, y, t) when z = and 0=0 when z = oo and when = 0, 
the appropriate solution is 



(B) 

In this case an element of the integrand corresponds to an instantaneous 
doublet whose direction is that of the axis of z. 

Let us next consider a case of steady heat conduction in a fluid moving 
vertically with constant velocity w. The fundamental equation is 

'N/J / CJ2/J O2/3 

(7(7 / O " \J " 

where K is the diffusivity. Writing = e"*/ 2 * the equation satisfied by is 

V 2 = A 2 0, 

* The transformation from one integral to the other can be made by successive differen- 
tiation and integration with respect to w of the firrt integral. 



344 Equations in Three Variables 

where A = W/ZK. A fundamental solution of this equation is given by 

= 



where R 2 = (x - ) 2 -f (y - 7?) 2 + (z - ) 2 , (& *?, constant). 

In particular, if = 77 = = 0, we have the solution = Ar~ l e~^ r , 
where r is the distance from the origin, and this corresponds to the solution 

6= Ar- l e* ( *- r \ ...... (T) 

This solution has been used by H. A. Wilson* and H. Machef to account 
for the following phenomenon. 

If a bead of easily fusible glass (0) be placed a few millimetres above 
the tip of the inner cone (K) of the flame of a Bunsen burner, a sharply 
defined yellow space (SS f ) of luminous sodium vapour is formed in the 

current of gas which is ascending vertically 
with considerable velocity. This space envelops 
the bead and broadens out in the higher part 
of the flame, as shown in Fig. 27. Provided 
the gas-pressure is not too high, the critical 
velocity of Osborne Reynolds, at which 
turbulence sets in, will not be exceeded even 
in these parts of the flame, so that the flow 
remains laminar, and the sodium vapour de- 
veloped from the bead is driven into the hot 
gas solely under the influence of diffusion. 

The fact that the vapour extends beneath 
the bead in the direction OA is proof of the 
high values of the coefficient of diffusion 
assumed at high temperatures, and at this 
point diffusion must be able to more than 
counteract the upward flow. Since an iso- 
thermal surface corresponds in the theory of diffusion to a surface of equal 
partial pressure, it is supposed that for suitable constant values of A and 
8 the equation (T) represents the surface enclosing the sodium vapour 
developed from the glass bead. When K is small and w large, this surface 
approximates to the form of a paraboloid of revolution with the origin as 
focus. 

Mache obtains the solution by integrating the effect of an instantaneous 
source which is successively at the different positions of a point moving 
relative to the medium with velocity w. In fact 




Fig. 27. 



j-O 

)-i 
Jo 



where A = w/2/c. 

* Phil. Mag. (6), vol. xxiv, p. 118 (1912); Proc. Camb. Phil Soc. vol. xu, p. 406 (1904). 
| Phil. Mag. (6), vol. XLVII, p. 724 (1924). 



Diffusion of Smoke 345 

A similar solution has been used by O. F. T. Roberts* to give the 
distribution of density in a smoke cloud when the smoke is produced 
continuously at one point, and at a constant rate. The case in which the 
smoke is produced continuously along a horizontal line at right angles to 
the direction of the wind is solved by integrating the solution for the 
previous case. 

5- 31. Two-dimensional motion of a viscous fluid. If (u, v) are the 
component velocities at the point (x, y) at time t, p the pressure at this 
point, the equations of motion, when the fluid is incompressible and of 
uniform density />, are 

du du du 1 dp 

+ u + v = -- _ 

ot ox oy p ox 

dv dv dv 1 dp _ 

+ u + v== -- _*: + v V 2 ?; , 

ot ox oy p oy 

while the equation of continuity is 



This last equation may be satisfied by writing 

ddf dJj 

<jj - ' ,j/ _ _ T_ 

U - <~\ ) ^ - ~f\~ j 

dx dy 

where if* is the stream-function, and if 

fccfc 

dx dy r 
is the vorticity at the point (x, y) at time t, we have 



or 



If s xv yu, we have 

ds dv du Ss dv du 
~- = x x -- v ^- + V J ^ = a; -^ -- y x~ 
9x dx y dx dy dy y dy 

d*s __ y*y _ d*u dv d*s _ 9 2 y 9 

9^ 2 - * 9^1 - y a^2 + 2 g^' 3^2 - x 

TT ^5 35 ds 1 / 3p 9^ 

Hence -^ + ^ 5 -+ v 5- = -- ( x -*r ~ 2/^" 
3^ 3a; 3y p \ dy y ox 

If x 2 + y 2 = r 2 we may write 

ds dv ds du 



ds ds f dv du\ 

u o~+ v 5" = M w a -- V-^~. 

3x 3t/ V 3r 9r/ 
* Proc. jRoy. Soc. London, vol. orv, p. 640 (1923). 



346 Equations in Three Variables 

If the flow is of such a nature that p depends only on r and v/u is 
independent of r, we have 



o- u 

since s = r v x , we have 

dr 



i 

. _r r / __ 
~ r 



Hence in the special case when ^ depends only on r, and the velocity 
is everywhere perpendicular to the radius from the origin, we have the 
d. ^rential equation 



This indicates that the velocity V = s/r satisfies the equation 



j*-Tj-fo- 

which is of the same form as the equation of the conduction of heat when 
the temperature is of the form = V cos 6. 

In the present case if/ and are related since they both depend on r 
and so the equation for is 



The equation satisfied by ^ is 



where / (t) is an arbitrary function of t. 
In the particular case when 



we have s - - (r 2 /2v^ 2 ) e- f2 / 4 *, F = - (r/ 

The total angular momentum is in this case 

TOO 

srdr = 



o 
and is constant. The kinetic energy is on the other hand 

TTO (^ V*rdr = Trp/2^ 2 . 
Jo 

This type of vortex motion has been discussed by G. I. Taylor* in 
connection with the decay of eddies. The corresponding type of vortex 
motion in which r _. j-i e -r/4v< 

ias been discussed by Oseenf, TerazawaJ and Levy. 

* Technical Report, Advisory Committee for Aeronautics, vol. I, 1918-19, p. 73. 

t C. W. Oseen, Arkiv f. Mat., Astr. o. Fys. Bd. vn (1911). 

J K. Terazawa, Report Aer, Res. Inst., Tokyo Imp. Univ. (1922). 

H. Levy, PhiL Mag. (7), vol. n, p. 844 (1926). 



Decaying and Growing Disturbances 347 

5-32. Solutions of the form iff = X (x, t) + Y (y, t). The condition to 
be satisfied is 



_ 
dy*dt dx lty dy dx* ~ 

Differentiating successively with respect to x and y we get 

d 2 X^Y_d*Yd*X^ 
dx* dy* dy* dx* 

We can satisfy this equation either by writing 



X = xa' (t) + b (t), Y = yA' (t) + B (t), ...... (B) 

, ... d'X r U ^&X d*Y r mi ,9 2 y /p . 

or by wnting -^ = [/t (f)] ^ , ^ = [ M (*)] ^ ....... (C) 

The supposition 

(D) 
...... ^ 



11^ A A 

leads to -^--- = 0, -~T = 0. 

3x 4 3^/ 4 

These equations follow from (C) if we put /u, (t) = 0. 
Solving equations (C) for X and 7 we get 

X = a(t) e**M + b (t) er* + xc (t) + d(t), 
Y - A (t) e^> 4- B (t) e-y*w + yC (t) + D (t). 

Substituting in the original equation and assuming that a (t), b (t), 
A (t), B (t) are not zero, we find that /x (t) must be a constant ^ and that 
the functions a, 6, c, A, B, G must satisfy the equations 

f - Cap,* = vap,*, ^b' -f 



primes denoting differentiations with respect to t. 

If the functions c (t) and C (t) are chosen arbitrarily, a(f), b (t), A (t) 
and B (t) may be determined by means of these equations when their 
initial values are given. In particular, if a = A = 0, c = C = 0, we can have 

6 - Pe 1 **, B = 



u = p,ye Vf *'* r ~ ILV 9 v = 
This represents a growing disturbance in which each velocity com- 
ponent is propagated like a plane wave. The pressure is given by the 
equation r/J/n 

F+J4_ + 1 



348 Equations in Three Variables 

The fluid may be supposed to occupy the region x > 0, y > 0. If so, 
fluid enters this region across the plane x = (Q > 0) and leaves it at the 
plane y = (P > 0). The amount entering the region is equal to the amount 
leaving the region if P = Q, the density p being assumed constant. 

If V = and p n is the pressure at infinity (a: = oo, y = oo), we have 
P - ^oo = pf ji*P*e 2 2vt -* (x+v) . 

The pressure is generally greater than p m and is propagated like a 
plane wave with velocity 

c == iiv A/2 = v - . 
^ u v 



Thus the velocity of a plane pressure wave in an incompressible fluid 
is equal to v times the ratio of the vorticity and the transverse component 
of velocity. 

When the motion is steady the equation to be satisfied is 
aer* (VIL -f C) + be-** (vji - C) + Aer* (vp> - c) + Be~ (vp, + c) = 0, 
and we have four typical solutions : 

= jps 2 + ex + qy* + Cy + D, 
ifj = v^y + be-** + ex + d, 
iff = VIJL (x + y) -f Ae, -f- be~* x + d, 
i/j = Aer* -f vpx -f Cy + D. 

^2 \r 

Returning to the first case we note that when -~ 2 = the equations 
(B) do not give all possible solutions, for if 

X - xa' (t) -f- 6 (0, 
the original equation becomes 



Writing C7 = -= 2 - we have the simpler equation 



which possesses a solution of type 

s U - f V"* 2 ' cos A [y - a (t)] o> (A) d\, 
Jo 

where o> (A) is a suitable arbitrary function. For the corresponding motion 

roo fJ\ 

4> = aw' () -f yc (*) + 6 (<) + c-*" cos \\y-a (t)} <a (A) -~, 

Jo A* 

r ^/A 

- c (t) - c-' sin A [y - a (<)] w (A) ^, 
Jo A 



v - 



Laminar Motion 349 

This solution may be used to study laminar motion. The corresponding 
solution for the case in which the motion is steady is 

*? 

i/i - Kx + Pe v + #7/ 2 + Ry + flf, 

where P,Q, It, S, K are arbitrary constants. If J -> while the coefficients 
P, Q, /?, $ become infinite in a suitable manner, a limiting form of the 
solution gives the well-known solution 

- Ay* + Qy* + Ry + S. 

It may be mentioned here that an attempt to find a stream-function iff 
depending on a parameter s but not on t, and such that 



1 ,14- *U * // X 

led to the equation ^ - ^ = / (a;, j,) 

The conditions for the compatibility of this equation and 

' 



seem to require / (#, y) to be a constant. By a suitable choice of axes the 
former equation may then be reduced to the form 



_ n 
dxdy~ ' 

and so = JT (x, s) 4- 7 ( 



EXAMPLES 

1. In the case when there is a radial velocity U and a transverse velocity F, both of 
which depend only on r and t and when the pressure p depends only on r and t, the equations 
for U and F are 



dV Tr dV UV (d 2 F 1 dV 1 T7 1 3 

a*' + ^ a~ ^ --- = v 1^~2 + ~ 3 ~ ~2 F T a" 
5^ dr r ( dr* r dr r 2 ) dr 

Hence show that F satisfies the equation 



F 
~ 



where ^L is a constant. If a X/2v, prove that there is a solution of type 

y ^ r 2<T+i t --2 e -r 2 /* v t 9 
and verify that the total angular momentum about the origin remains constant. 

2. Prove that the equation for F is satisfied by a series of type 

y _ r n*-m 1 1 __ _ . , _ . ____ (r 2 /vt} 
V rt \ (1 + - 2a) (n + 3) (T /vl> 

_ 
v ' ; ' 



350 Equations in Three Variables 

and verify that when n = 1, m = 2, 



rf-'e- <" (l + --"- (r*/M) + -^ 1 (rV4rf)* + ...1. 

^ CT I (7 & 41 J 



This is a particular case of Kummer's identity F (a; y; a;)e" a: = ^(y a; y; ~^), where 
F (a; y; ar) is the confluent hypergeometric function (Ch. ix). 

3. Prove that there is a type of two-dimensional flow in which 

{ = *V, 
and is consequently of the form 

t = -**** F (x, y), 
where F (x, y) satisfies the differential equation 



A 
= 0. 



Prove that in the latter case if a 2 > 6 2 there is a growing disturbance which is propagated 
with velocity vk 2 /a, and show that 



Discuss the cases in which 

F = cos ax cos 



a u' 



CHAPTER VI 

POLAR CO-ORDINATES 

6-11. The elementary solutions. If we make the transformation 

x = r sin cos <, y = r sin sin <, 2 = r cos 0, 
the wave-equation becomes 
&W 2dW 1 3 



/ . 3]f\ _! __ _ 

"3r r ~3r " r 2 sin 30 \ sm 30 ) + r 2 sin 2 9e/> 2 c 2 9* 2 ~ 
This is satisfied by a product of type 

W=R(r)(0)Q(<f>)T(t), ...... (I) 

Mm 
if + WcT = 0, 



sin i 



ar* r ar 

where k, m and n are constants. 
The first equation is satisfied by 

T = a cos (to) 4- 6 sin 
where a and 6 are arbitrary constants ; the second equation is satisfied by 

O = A cos m<f) 4- J5 sin m<, 

where .4 and B are arbitrary constants. The third equation is reduced by 
the substitution cos = p, to the form 



Its solution can be expressed in terms of the associated Legendre 
functions P n m (/x) and Q n m (p) which will be defined presently. 

When k = the fourth equation has the two independent solutions r n 
and r~ (n+1) , except in the special case when n = (n 4- 1), i.e. when 
n = . Making the substitution w = r% R in this case we obtain the 

equation 

d*w 1 dw _ 
dr^^r dr ~ U ' 

which is satisfied by w == C + I> log r, where (7 and Z> are arbitrary 
constants. 



352 Polar Co-ordinates 

The fact that r n and r" n ~ l are solutions of the equation for R furnishes 
us with an illustration of Kelvin's theorem that if / (x, y, z) is a solution 
of Laplace's equation, then 

1 f ( x y, ?\ 

r J \r*> r 2 ' r*) 

is also a solution. The transformation in fact transforms r n 0<1> into r~ n ~ l 0O ; 
it also transforms r~i (G -f D log r) Q<J> into r~* (G D log r) 0<I>. 

When m = and n = the differential equation for is satisfied by 



Thus, in addition to the potential functions 1 and -, we have the 

potential functions 

1 ! r + z ill r + z 
2 lo 8 r _i and 2r log F-V 

It should be noticed that 

3 /I. r + z\ 1 



1 1 

In fact we have ^- log (r + z) = - , 



and it is easily verified that log (r -f z) and log (r z) are solutions of 
Laplace's equation. These formulae are all illustrations of the theorem that 
if W is a solution of Laplace's equation (or of the wave-equation), then 



is also a solution of Laplace's equation (or of the wave-equation). 

6-12. In the case of the wave-equation the solution corresponding to 
l/r is e tkr /r, and there are associated wave-functions 

- cos k (r ct), - sin k (r ct), 
which are, of course, particular cases of the wave-function 



in which /(T) is an arbitrary function which is continuous (D, 2). 

6-13. In the case of the conduction of heat the fundamental equation 
possesses solutions of the form (I) where R, 0, O satisfy the same 
differential equations as before but T is of type 

a exp (- 



Cooling of a Spherical Solid 353 

where a is a constant and h 2 is the diffusivity. Thus there are solutions 
of type 



- cos kr . e~w , - sin kr . e- 



which depend only on r and t. The second of these is the one suitable for 
the solution of problems relating to a solid sphere. If, in particular, there 
is heat generated at a uniform rate in the interior of the sphere the 
differential equation for the temperature 6 is 



ot 
where 6 is a constant. There is now a particular integral b*r 2 /6h 2 which 

r)f) 

must be added to a solution of -^ = h 2 V 2 9. 

ct 

If initially 6 *= throughout the sphere, 6 being a constant, and the 
boundary r = a is suddenly maintained at temperature 1 from the time 
t = to a sufficiently great time T, the condition at the surface is satisfied 
by writing 



while the initial condition is satisfied by writing* 




_ 

As -> oo, tends to the value t + - . 2 -- ; and ^- to the value 

6 2 r 
^r-, so that the flow of heat across the surface is, per second, 



O K 

Writing 6 2 = and h 2 = , where p is the density and a the specific 

heat of the substance, we have the result that the rate of flow of heat 
across the surface is 4Q7ra 3 /3, a result to be anticipated. 



_ 

If, on the other hand, the initial temperature is t -\ -- 7^2 

the surface of the sphere radiates heat to a surrounding medium at 
temperature 2 at a rate E (9 a 2 ) per square centimetre, where 6 a is the 
(variable) surface temperature of the surface of the sphere, the solution is 



e - B - ~ + - XD n e~ n2 M sin nr. 
on* r 

* The constant D m is obtained by Fourier's rule from the expansion of - 6 l - ^ 2 (a 2 - r a ) 
in a sine series. 

B 2 



354 Po/ar Co-ordinates 

The surface condition is satisfied by writing 

R - fl 4- a6 * (^ + 

2 + 



where <p m is the with root of the transcendental equation 



The initial condition gives 



Fr = D n sin nr, 

m-l 



where 

and the extended form of Fourier's rule gives 

z- K)' 
Ea - E 

(Ea - 



2F K 2 <l> m 2 4- (Ea - K}* f . (r^ m \ 3 
I~ T ~W~TF W\ r - sm I ) d r 



These results have been used by J. H. Awbery* in a discussion of the 
cooling of apples when in cold storage. 

6-21. Legendre functions. The method of differentiation will now be 
used to derive new solutions of Laplace's equation from the fundamental 

i , . 1 , 1 , r + z 
solutions - and ^- log - . 
r 2r & r - z 

After differentiating n times with respect to z the new functions are of 
form r- n ~ l Q; consequently we write 



n 



i Lt 2 
g - 



'' n ^>- n\ dz\2r^r-z) 
and we shall adopt these equations as definitions of the functions P n (p.) 
and Q n (p) for the case when n is a positive integer and 6 is a real angle. 
The first equation indicates that there is an expansion of type 

(^2 tynfii I /Tf2\ j \* - ~P ( ii\ I n I *?* I Y 
^ti-/ fJi i~ i4/i ^j />w+l * n \r^)i I ^^ I 

n 

and this equation may be used to obtain various expansions for P n (fi). Thus 
1.3 ... (2n- 1) 



3 ^= 1.2..., 



V- + ...] 



- ~ = (-)- F - n, n 



. (7), vol. iv, p. 629 (1927). 



Hobsorfs Theorem 355 

where F (a, 6; c; x) denotes the hypergeometric series 

1 + a ' b x . MJi!L 6 J*JQ *;2 , 
+ l.c* 4 " f.2.c(c + 1) "" 

6-22. Hobsorfs theorem. The first expansion for P n (^) is a particular 
case of a general expansion given by E. W. Hobson*. If / (x, y, z) is a 
homogeneous polynomial of the nth degree in x, y, z, 



r 2 V 2 r*V 4 

X 



2 (2?) 2.4 
When / (x, y, z) = z n this becomes 



1 , 
^3) '"J ; ( * J y ' Z) ' 



r n (n - 1) 2 n ( -J)Jn -_2) Jn -3) 4 _ 1 

A 1 Z 2 (2/1 - : 1) z + 274 (271 - 1) (2n - 3) "'J' 

which is equivalent to the expansion for n I r~ n ~ l P n (^). 

Assuming that the theorem is true for / (x, y, z) = z n it is easy to see 
that the theorem must -also be true for / (x, y, z} = (x -f r^y + ^) n , where 
|x + 77^ + ^ is derived from z by a transformation of rectangular axes, for 

r\ /"\ ^\ ^\ 

such a transformation transforms ^- into ^ ^ -f -n ^- + t - and leaves 

S^ ra ' cy dz 

V 2 unaltered. 

To prove that the theorem is true in general it is only necessary to 
show that / (x, y, z) can be expressed in the form 

/ (x 9 y,z)= 2 A s ($ 8 x + w + ,*), 

s=l 

where the coefficients A s are constants. 

To determine such a relation we choose k points such that they do not 
all lie on a curve of degree n and such that a curve of degree n can be drawn 
through the remaining k I points when any one of the group of k points 
is omitted. Let s , rj s , s be proportional to the homogeneous co-ordinates 
of the sth point and let i/j s (x, y, z) = be the equation of the curve of 
degree n which passes through the remaining k 1 points. 

Assuming that a relation of the desired type exists we operate on both 

sides of the equation with the operator i/j 8 ( ^ , ^ , ^ - j . The result is 

' ~dy' 

Giving s the values 1, 2, ... k all the coefficients are determined. Since 
a curve of the nth degree can be drawn through \n (n -f- 3) arbitrary points, 

* Proc. Land. Math. Soc. (1), vol. xxiv, p. 55 (1892-3). 

23-2 



356 Polar Co-ordinates 

the number k should be taken to be \ (n + 1) (n + 2), which is exactly the 
number of terms in the. general homogeneous polynomial / (#, y, z) of 
degree n. The coefficients A 3 could, of course, be obtained by equating 
coefficients of the different products x a y b z c and solving the resulting linear 
equations, but it is not evident a priori that the determinant of this system 
of linear equations is different from zero. The foregoing argument shows 
that with our special choice of the quantities g s , 77 5 , 8 the determinant is 
indeed different from zero because with a special choice of /, say 



the equations can be solved. 

The solution is, moreover, unique because if there were an identical 
relation 

= S C s (t s x + w + ,*), 

3=1 

the foregoing argument would give 

O^nlC^tf,,^, .). 

Hobson's theorem has been generalised so as to be applicable to 
Laplace's equation for a Euclidean space of m dimensions. Writing 

V . = - + ^+ + ** 
m -^^ " [ ' 



and using/ (^ , a; 2 , ... x m ) to denote a homogeneous polynomial of degree n, 
the. general relation is 

\' 4' "' 9 



[r' 2 V m 2 
1 ~ 2 (m + 2n - 4) + ^ 



2n - 6) 

6-23. Potential functions of degree zero. When n = the differential 
equation satisfied by the product U = 0O may be written in the form 

a 2 17 



_ 

~ ' 

U f ^ f <W 1 -4. ^ 

where s = - -, ^ 9 = ~^ 5 ^ log.tan - . 

) I ~ p. 2 ] smO &i 2 

It follows that there are solutions of type 



where /is an arbitrary function and/ (u) = 
This solution may be written in the form 



Differentiation of Primitive Solutions 357 

where F is an arbitrary function. The general solution of Laplace's equation 
of degree zero may thus be written in the form 



r 



where F and G are arbitrary functions*. The general solution of degree 1 
may be obtained from this by inversion and is 

\ (x ii 



- 



>r + zj r \r + z. 

Solutions of degree (n 4- 1) may be obtained from the last solution 
by differentiation. In particular, there is a potential function of type 



^ H ix + i 
~~ 8z [r\z + r 



which is of the form r-"- 1 ^ (0) e m *. The function must consequently be 
expressible in terms of Legendre functions. When m is a positive integer 
equal to or less than n we have in fact the formula of Hobson 



) n n /T .L ?A m n 

(/ (ITT) J = { ~ }n (n ~ m) ! r " n " lp 



When m is a positive integer greater than or equal to n we have the 
expansion 

fl ^ (9-n 1\ 1 ** (9n ^\ 9m 1 
, . \ L . O . . . \LTl 1) L . o . . . I 4 /I *> l i 71 1 , . 

: V ) I ~2V~1 / rn^n ~^ ~2n~? m~n I 1 ( m ~~ n ) 

+ r 2n-I7^ r Tm-nV2 ~~ j 2 (w - 7l) (m - ?* + 1) + ... 

... 4- _j (m n) (m n+ 1) ... (m 

which may be used to define the function x (0) in this case. In particular, 
we have the relation 

an _ r ___ ! 

dz n [r (z -f r 

When this is used to transform the expression for r~ n - l P n " 
we find that| 



* W. F. Donkin, Phil. Trans. (1857). 

t This formula is given substantially by B. W. Hobson, Proc. London Math. Soc. (1), vol. xxn, 
p. 442 (1891). Some other expressions for the Legendre functions are given by Hobson in the 
article on "Spherical Harmonics" in the Encyclopedia Britannica, llth edition. 



358 Polar Co-ordinates 

EXAMPLE 

Prove that if m is a positive integer 



m pi 
?r J 

T 2 
/ 

T J o 



, e lma da 

z + ix cos a -f ii/ sin a r \z + r 

\ ( x 



z 



i , . . . v , 

log (z -f ix cos a + it/ sin a) e tma da = - - r ~~ 

-\- r / 



6-24. Upper and lower bounds for the function P n (/x). We shall now 
show that when 1 < /x < 1 the function P n (/x) lies between 1 and -f 1. 
This may be proved with the aid of the expansion 

--" cos n>8 -f- A . -~ - * - - cos (n 2) 6 

1.3 1.3 ... (2n- 5) 

+ n 



2.4-2.4... (2n-4r wv " ' ' ' 
which is obtained by writing 

(1 - 2x cos 9 -f x 2 )~i - (1 - xe*)-* (1 - ze-'T** 

and expanding each factor in ascending powers of x by the binomial 
theorem, assuming that | x \ < 1. 

It should be observed that each coefficient in the expansion is positive, 
consequently P n (/*) has its greatest value when 6=0 and /x, = 1, for then 
each cosine is unity. 

If, on the other hand, we replace each cosine by 1, we obtain a 
quantity which is certainly not greater than P n (/x). Hence we have the 
inequality - i < P. (/t > < i, fc* _ 1 < ,. < 1. 

When n is an odd integer P n (/x) takes all values between 1 and + 1, 
but when n is an even integer P n (ju) has a minimum value which is not 
equal to 1. This minimum value is | for P 2 (p) and f for P 4 (/x). 

6-25. Expressions for the Legendre polynomials as nth derivatives. 
Lagrange's expansion theorem tells us that if 

z =-- IJL + acf) (z), 

the Taylor expansion of/' (z) -v- in powers of a is of type 

(tjJL 



az = - - 



1 . 

a 2 )*, - = (1 - 2/xa + a 2 )-*; 



Formulae of Rodrigues and Conivay 359 

a comparison of coefficients in this expansion and the expansion 

(1 - 2/ta-f a 2 )-* = S aP n ( M ) 
o 

gives us the formula of Rodrigues, 

p ^ - 2 -4l - [( " 3 ~ 1)nj - 

If, on the other hand, we write 

c (z) - 2 (V~z - 0, 
we have 3 = /z 4- 2a (Vz t) 9 

z - 2a Vz + a 2 = a 2 - 2a^ + ti, 



= a a 2 2a^ -f M, 

n + 1 






/ M = ?" _ 3 1 

\vV/ n ! 3/x n vV 
1 c n (r t) n 



Hence 



or ....- ,-t * i :. i ~\( r dr) n r 

This formula is due to A. W. Conway, the previous one to E. Laguerre. 
Replacing t by z we have the following expression for a zonal harmonic 

1 , . 1 3 n (^-^Y 

r ' 



z and r being regarded as independent. 

6-26. The associated Legendre functions. The differential equation (II) 

m 

of 6-11 is transformed by the substitution = (1 p, 2 ) 2 P to the form 



but this equation is satisfied by P = j-~ , where v is a solution of Legendre's 

equation 

, // 2 7) fin 

(I-/**) ^,-2/*;fc+<+l)'-0, 
particular solutions of which are P n (/x) and Q n (/x). 



360 Polar Co-ordinates 

Hence we adopt as our definitions of the functions P n m (/z) and Q n (/x) 
for positive integral values of n and m 



m 

o n m 



- 1 < /x< 1. 

With the aid of these equations we may obtain the difference equations 
satisfied by P n m (p.) and Q n m (//,) : 

(TI - m 4- 1) P w r?+1 - (2n 4- 1) /zP n + (n + m) P m n ^ = 0, 



+ l = 2mp.P n m - (n + r/i) (n - m + 1) Vl 
P"*,,., - jzP n - (n - m + 1) Vl - 
P m n+1 - /xP w w 4- (n 4- w 



1 - (n 4- m 4- 1) fi.P n m - (n - m 4- 1) P^+i, 
and the following expressions for the derivative 

(I-/*') ^ P n (/,) - (n + 1) /JV (/z) - (^ - m + 1) P- n ,! (/x) 
- (w + w) P- w _i (/x) - n/iP n ~ (/x). 

Similar expressions hold for the derivative of Qn w (/*) 

Expressions for the Legendre functions of different order and degree n 
are easily obtained from the difference equations or from the original 
definitions. In particular 

P = 1- 

P! - cos 0, P^ - sin 6. 

P 2 = i (3 cos 2 19-1), Pa 1 - 3 sin 9 cos 0, P 2 2 - 3 sin 2 5. 

P 3 - \ (5 cos 3 0-3 cos 0), P3 1 - | sin (15 cos 2 - 3), 

P 3 2 - 15 sin 2 9 cos 9, P 3 3 - 15 sin 3 0. 

P 4 - I (35 cos 4 0-30 cos 2 9 4- 3), P^ - J sin (35 cos 3 0-15 cos 0), 

P 4 2 - sin 2 (105 cos 3 - 15), P 4 3 - 105 sin 3 cos 0, P 4 4 - 105 sin 4 0. 

P 6 o = j (63 cos 5 0-70 cos 3 04-15 cos 0), 

Pg 1 - I sin (315 cos 4 - 210 cos 2 4- 15), 

P 6 2 - \ sin 2 (315 cos 3 - 105 cos 0), 

P 6 3 ^ | S i n 3 ^ (945 C os 2 _ 105), 

P 6 4 - 945 sin 4 cos 0, P 5 5 - 945 sin 5 0. 



A ssociated Legendre Functions 361 



EXAMPLES 
1. Prove that if m and n are positive integers 



2. Prove that 



ai ~~ * cf") ^ Pn?W (/i) 6 * m ^ = ( + m) (w + w - 1) r n ~ l P n 
(jx + i d] [r "" n "" 1 Pr * m (fz) ^^ = "" r ~ n ~ 2 Pn + lWl 

(w ~ m + 1J (n ~ 



6-27. Extensions of the formulae of Rodrigues and Conway. By 
differentiating the formula of Rodrigues m times with respect to ^ we 
obtain the formula 



We shall use a similar definition for negative integral values of m and 
shall write 



Expanding by Leibnitz's theorem we obtain 



n m ( n __ w\ f 72, 1 

_ V V Af> AA6 ^ ' __ _ 

~ 



Comparing the two series, we obtain the relation of Rodrigues 



362 Polar Co-ordinates 

This may be derived also from the equations of Schende 



/ \m /I i ,,\ 2 // 

= ( ~> _ (~-^i - rfu - n n+ 

2(n-m)!\l- p.) d^ 1 ^ > 



~- 

(n-m)!\l- p. 

which may likewise be proved with the aid of Leibnitz's theorem. We 
have in fact 

^[(/x-ir-^+i)^] 

nl (n-m)\ (nm] ! _ s 

" ^ *T(^^ ^-^T^)"! (m + ) ! (/A ' ( ^ + j ' 
By differentiating Conway's formula m times with respect to t and 

m 

multiplying by (r 2 2 ) 2 we obtain the formula 

" ' 0. 



. - ,, 

r n+i y r y v ; v ' (n m)\ \rdrj r 

Making use of the formula (C) we may also write 



Changing the sign of m we have 



This formula also holds for m > 0. 

6-28. Integral relations. The Legendre functions satisfy some interesting 
integral relations which may be found as follows : 

Writing down the differential equations satisfied by P n m (/x) and P t k (/z) 



let us first put k = m and multiply these equations respectively by Pf 
and P n m and subtract, we then find that 



+ (n - I) (n + I + 1) P n m P^ = 0. 



Integral Relations 363 

Integrating between 1 and + 1 the first term vanishes on account of 
the factor 1 /x 2 and so we find that if I ^ n 



Next, if we put I = n and multiply by P n fc , P n m respectively and 
subtract we find in a similar way that if ra 2 ^ k 2 



To find the values of the integrals in the cases I = n, k = m we may 
proceed as follows : 

If we multiply the first difference equation by P m n+1 (/x) and integrate 
between 1 and + 1 we obtain the relation 

(n - m + 1) f 1 [P*' B+1 ( M )] 2 dp - (2n + 1) P iiP m n+l P n m rf/x, 
J-i J-i 

while if we multiply it by P m n _! (/x) and integrate we obtain the relation 

(n + m) f 1 [P- n _ x Oz)] 2 ^ = (2n -f 1) f l /zP^P'V^. 
J-i J-i 

Changing ?i into n 1 in the previous relation we find that 

(2n + 1) (n - m) [' [P w (^)] 2 rf^ = (2w - 1) (n + m) f [P- n _ t (/x)] 2 rf/i. 
J-i J-i 

But 



Therefore 

[P n m (/*)? ^= l-3 ... (2m - 1)* j^l - M ) dp = ^-j (2m) ! 

and so f ' [P.* (^ ^ = 9 -^rT r ^ ' 

]-i r/J r 2n+l(n~m)\ 

Let us next multiply the difference equations 



f 

(1 - M 2 ) ^ = n^P^ -(n- m) P n - 

by (1 ijP)- l P m n ^ and (1 /x 2 )~ 1 P w w respectively and add. 
Integrating between 1 and -i- 1 we obtain the relation 



(n + m) [P-W]* - - (n - m) 

= if m > 0. 



364 Polar Co-ordinates 

Now 

f 1 [P n m (^)]\4r~^ I 2 .3 2 ...(2m- I) 2 f 1 (I-/* 2 )" 
= 2. (2m- 1)!. 

MIL. r f 1 rr , / xno ^M- 1 (?l -f m) ! 

Therefore [P n m (M)] _ 2 = . V- - f . 

These relations are of great importance in the theory of expansions in 
series of Legendre functions. See Appendix, Note m. 

6 29. Properties of the Legendre coefficients. If the function / (x) is 
integrable in the interval 1 < x < 1, which we shall denote by the 

symbol /, the quantities 

/> 1 

(3*} doc (I) 

are called the Legendre constants. If these constants are known for all 
the above specified values of n and certain restrictions are laid on the 
function/(x) this function is determined uniquely by its constants. An 
important case in which the function is unique is that in which the function 
(I x 2 )%f(x) is" continuous throughout /. To prove this we shall show 
that if <f) (x) = (I # 2 )~i</r (x), where iff (x) is continuous in /, then the 
equations 

f 1 <f>(z)P n (x)dx=0 (n= 0,1,2,...) (II) 

j-i 

imply that $ (x) = 0. 

The first step is to deduce from the relations (II) that 

( <f>(x)x n dx= (TI= 0, 1,2, ...). 

This step is simple because x n can be represented as a linear com- 
bination of the polynomials P (x), P x (x), ... P n (x). 

The theorem to be proved is now very similar to one first proved by 
Lerch*. The following proof is due to M. H. Stonef. 

If ^ (x) ^ for a value x = in / we may, without loss of generality, 
assume that $ () > 0, and we may determine a neighbourhood of 
throughout which <f> (x) > m > 0. Now if A > the polynomial 

p (x) *== A %A (x ) 2 (x 2 -f 1) 

is not negative in / and has a single maximum at x == . We choose the 
constant A so that in the above-mentioned neighbourhood of' f there are 

* Acta Math. vol. xxvn (1903). 

f Annals of Math. vol. xxvii, p. 315 (1926). 



Theorems of Lerch and Stone 365 

two distinct roots of the equation p (x) = 1 which we denote by x l , x 2 , the 
latter root being the greater. We thus have the inequalities 
0< p (x) < 1, - 1 < x< x ly 

X 2 < X < 1, 

p (x) > 1, <f> (x) > m, x l < x < x 2y 
ifj (x) > - M , - Kx< I, 

w 

where M is a positive quantity such that M is a lower bound for the 
continuous function (x). 

Writing p n (x) [p (#)] n , we have 

<t>(x)p n (x)dx=0, n=l,2. (A) 



L 



On the other hand 



fx, rx, 

< (*0 Pn (x) dx> m\ p n (x) dx, 

Jxi Jx l 

f ' <f>(x)p n (x)dx> - M\ l (1 -x*)-ldx, 
J-i J-i 

r 1 f 1 

<f> (x) p n (x) dx > - M\ (I- a; 2 )-* dx, 

J x, J x t 

r 1 ( x * * r 1 

^ (^) Pn ( x ) dx> m \ p n (x) dx M \ (1 # 2 

J-l J Xi J-l 

f* 1 
> m p n (a?) rfa; rrM. 



fx, 

Since p n (x) dx -> oo as n -> oo we can choose a number N such 

that the right-hand side is positive for n > N. This contradicts (A) and so 
we must conclude that <f> (x) = throughout /. 

Lerch's theorem is that if ijj (x) is a real continuous function and 

/i 

x n (x) dx = for tt = 0, 1, 2, ... to oo, then iff (x) = 0. 
Jo 

By Weierstrass's theorem the function i/j (x) may be approximated 
uniformly throughout the interval (0, 1) by a polynomial G (x). In other 
words, a polynomial G (x) can be chosen so that ifj (x) = G (x) -f 86 (x), 
where | (x) \ < 1, and 8 is any small positive number chosen in advance. 
Now if t/j (x) is not zero throughout the interval (0, 1) we can choose our 

number 8 so that 

Ji ri 

o Jo 

But, since G (x) is a polynomial, we have 



366 Polar Co-ordinates 

Therefore f V (*) [<A (*) ~ 8 

Jo 

or P[<A (a)] 2 da = s[ 9 (x) (a) 

J o Jo 



f 1 

Jo 



[ 
o >o 

This contradicts (B) and so we must have (x) = 0. Putting x = e~* 

roo 

we deduce that if er* $ (t) dt = f or 2 > 0, and </> () is continuous for 

Jo 
t > 0, then f/> (J) = 0. 

EXAMPLES 
1. When m and n have positive real paits 



A Q m (z) Qn (2) dz - *(n -hi)- *(m -h 1) - 2 l 7r [( - T) sin (\m -f- Jw) 



where ^(z) = log r (2), 

A = (m - rt) (m -f n-f I) 

H = /*(}w + J, J'" + 1) j ; , r 

[S. (* Dharand \ (1 Shabde, Hull Calcutta Math Soc. v. 24, 177-186 (1932). 



also that v\ith the same notation 

Q m (z) Qn (2) dz - t (m + 1) - * (w -f 1) 
((^ne.sh Prasad, Proc Henarex Math. Soc. v 12, pp. 33-42, 19.) 



2. Show by means of the relation 



that when n is a positive integer the equation P n (ft) = has n distinct roots which all lie 
in the interval 1< /x< 1. 

3. Prove that when m and n are positive integers 

2,m+n-\i <( m . W \u4 

M4- 2 ^ m *- n P ^^P (2^^ - .i 7 ^^ 71 ;-!. 

i + 2 ) r rn (.) r w (zj a~ - {mlnl}2 (2m + 2n 4 _ X) ,. 

[E. C. Titchmarsh. 

An elementary proof of this formula is given by R. 0. Cooke, Proc. London Math. Soc. (2), 
vol. xxin (1925); Records of Proceedings, p. xix. 



Green's Function for a Sphere 367 

6-31. Potential function with assigned values on a spherical surface S. 
Let P, P' be two inverse points with respect to a sphere of radius a. 
If is the centre of the sphere we have then 

OP. OP' -a 2 , 

and 0, P, P' lie on a line. The point is sometimes called the centre of 
inversion. 

If P lies inside the sphere, P' lies outside ; if P is outside the sphere, P' is 
inside. If P is on the sphere, P' coin- 
cides with P. If P describes a curve 
or surface P' will describe the inverse 
curve or surface and it is clear that 
a curve or surface will intersect the 
sphere at points where it meets its 
inverse. If a curve or surface inverts 
into itself it must intersect the sphere 
S orthogonally at the points where it 
meets it because at these points two 
coi sccutive inverse points lie on the 
surface and on a line through 0. This line is then a tangent to the surface 
and a normal to 8 at the game point. If M s is any point on S the triangles 
0PM s , OM S P' are similar, and we have 

OP 




PM 8 ~P'M,' 

If charges proportional to OP and OM S are placed at P and P 1 
respectively, the sum of their potentials at any point M s on S will be zero. 
Writing OP =_r, PM =-- R, P'M - R' ', where I/ is any point, we see that 
the function 

__ I a 1 
^ PM ~P"r*^ 

is zero when M is on S and is infinite like - at the point P. We shall call 

Zi 

this function the Green's function for the sphere. G P}J is easily seen to be 
a symmetric function of the co-ordinates of P and M, r if JT is the 
inverse of Jf we have 

Oif OP 
P'M ^ PM 7 ' 

The point P' is called the electrical image of P and (7 />3/ represents the 
potential at M when the sphere $, regarded as a conducting surface at 
zero potential, is influenced by a unit charge at P. When a becomes 
infinite and recedes to infinity the sphere becomes a plane, P' is then the 
optical image of P in this plane, and the virtual charge at P' is equal and 
opposite to that at P. 



368 Polar Co-ordinates 

Now let OM = r' and POM = <*>, then 



2 __ 2rr cos co, 
__ 2r' cos co, 



a a ( a i + r 2 - 2ar cos a>)' 

Let (r, 0, <), (r, 0', </>') be the spherical polar co-ordinates of the point 
P and a point M s on the surface of S, then the theorem of 2-32 tells us 
that if a potential function V is known to have the value F (#', <') at a 
point M 8 on $ then an expression for V suitable for the space outside S is 



o (a 2 -f r 2 - 2ar cos co) 

while a corresponding expression suitable for the space inside S is* 

1 fir r2n a (a 2 - r*)F (#',</>') sin 6' 
V (r, 9, (/>) - -i- rffl' <Z<A' - ~ 7 ~~7 1^ ^T" - 
^ ' ^ ; 47rJo Jo (a 2 + r 2 - 2ar cos w)* 

When the sphere becomes a plane the corresponding expression is 

, , /(',') 



the upper or lower sign being taken according as z ^ 0. In this case 

/ (x 1 ', y') = V (x f ', ?/', 0) is the value of F on the plane 2=0. 

6' 32. Derivation of Poissons formula from Gauss's mean value theorem. 
Poisson's formula may also be obtained by inversion, using the method of 
Bocher. 

Let us take P' as centre of inversion and invert the sphere 8 into itself. 

The radius of inversion is then c = (r 2 a 2 )* = - (a 2 r 2 )i, where c is the 

length of the tangent from P' to the sphere, it is real when P' is outside 
the sphere and imaginary when P' is within the sphere. (In Fig. 28 
OP'-r .) 

Let Q, Q' be two corresponding points on $, then the relation between 
corresponding elements of area is 

cr \ 4 



Writing dS' = a*dl', dS = a 2 dfl, where d&' and dO are elementary- 
solid angles, we have 

dtt = ( C p^) dl = (a 2 - r 2 ) 2 (r 2 + a 2 - 2ar cos w)- 2 dQ, 

\U- . x v^/ 

where co is the angle between OQ and OP. 

* This is generally called "Poissoi^s integral," both formulae having been proved by S.D. 
Poisson, Journ. cole Polyt. vol. xix (1823). The formula for the interior of the sphere had, how- 
ever, been given previously by J. L. Lagrange, ibid. vol. xv (1809). 



Poisson's Formula and its Generalisations 369 

Now if V 1 ' Q > is a potential function when expressed in terms of the 
co-ordinates of Q', the function 



P'Q Q ' 

is a potential function when expressed in terms of the co-ordinates of Q, 
consequently the mean value theorem 



glV6S 



47rF ' = a (a 2 - r 2 ) 2 f F dQ [r 2 + a 2 - 2ar cos ]*, 

CI* J 



and since c. F ' = P'P. V P , we have crV ' = (a 2 r 2 ) F P , and our formula 
is the same as that derived from the theory of the Green's function. 

This method is easily extended to the case of hyperspheres in a space 
of n dimensions. The relation between the contents of corresponding 
elements of the hyperspheres is now 

*?_' _ f?'^\ n ~ l - (JLV n ~ 2 - ( cr V w ~ 2 
dS-\P'Q) ~(P'QJ ~~\a.PQ> ' 

while the relation between corresponding potentials is 



Writing the mean value theorem in the form 



the generalised formula of Poisson is 



cr 



n 
*2\n-2 f /rr\n f jr 

- ) F P US' = ( - J F dS [r 2 + a 2 - 2ar cos a>] 2 
/ J VQ- / J 



71 

or V P J AS' = a"- 2 (a 2 - r 2 ) J F Q cZS [r 2 + a 2 - 2ar cos cu] " ^. 

6*33. /Some applications of Gauss's mean value theorem. The mean value 
theorem may be used to obtain some interesting properties of potential 
functions. 

In the first place, if a function F is harmonic in a region .R it can have 
neither a maximum nor a minimum in R. 

If the contrary were true and F did have a maximum or minimum 
value at a point P of R the mean value of F over a small sphere with 
centre at P would not be equal to the value at P. If now the sphere is made 
so small that it lies entirely within jR, Gauss's theorem may be applied 

B 24 



370 Polar Co-ordinates 

and we arrive at a contradiction. Since a function which is continuous 
over a region consisting of a closed set of points has finite upper and lower 
bounds which are actually attained, we have the theorem : 

// a Junction is harmonic in a region R with boundary B and is con- 
tinuous in the domain R -|- B, the greatest and least values of V in the domain 
R |- n are attained on the boundary B. 

One immediate consequence of the last theorem is that if the function 
V is harmonic in R, continuous in R -f B and constant on B it is constant 
on R -f B. This theorem is important in electrostatics because it tells us 
that the potential is constant throughout the interior of a closed hollow 
conductor if it is known to be constant on the interior surface of the 
conductor. Another interesting consequence of the theorem is that if the 
function V is harmonic in R, continuous in R \ B and positive on B it is 
positive in R 4- B. For if it were zero or negative at some point of R the 
least value of V in R would not be attained on the boundary*. 

This theorem may be restated as follows: 

If \\ and V 2 be functions harmonic in R and continuous in R 4- B, 
and if V v is greater than (equal to or less than) V 2 at every point of B, 
then \\ is greater than (equal to or less than) V 2 at every point of E -f B. 

A converse of Gauss's theorem, due to Koebe, is given in Kellogg's 
Foundations of Potential Theory, p. 224. 

6' 34. The expansion of a potential function in a series of spherical 
harmonics. If V (x, y, z) is a potential function which is continuous 
throughout the interior of a sphere S and on its boundary, and whose 

first derivatives ^ , ^ , ^ arc likewise continuous and the second 

ex en cz 

derivatives finite and integrable (for simplicity we shall suppose them to 
be continuous) then V admits of a representation by means of Poisson's 
formula and it will be shown that V can be expanded in a convergent 
power series in the co-ordinates x, y, z relative to the centre of S. Writing 

cos o cos 6 cos 6' -f sin 8 sin 6' cos (</> <') //,, 
we have 

2 /2\ oo n l 

P. ( M ) | r | > a, 



(a 2 4- r 2 - 2 



., - - 

(a 1 -f r 1 -- 2<7rcosoo)* n 

Substituting in the expressions for V we may integrate term by terrr. 
because the series are absolutely and uniformly convergent on account ot 
the inequality | P n (p) \ < 1. 

* See a paper by G. E. Ray nor, Annals of Math. (2), vol. xxm, p. 183 (1923). 



Expansion of a Potential Function 371 

We thus obtain the expansions 



n=0 " 



7- S ( 

n=0 

where in each case 



S n (6, ft = I \' f 'V (0', 

47Tj() JO 



sn 



The function r n S n (0, <f>) is called a,spherical harmonic or solid harmonic 
of degree n, it is a polynomial of the nth degree in x, ?/, z arid is a solution 
of Laplace's equation because r n P n (/*) is a solution. 

The function S n (9, <f>) is called a surface harmonic, it may be expressed 
in terms of elementary products of type P n m (cos 0) e* tn * by expanding 
P n (fji) in a Fourier series of type 

P n (/A) = 2 F n m (0,0')e" <+-*'>. 

in n 

By expanding r n P n (p) in a series of this form and substituting in 
Laplace's equation (in polar co-ordinates) we get a series of typeSC f w e* m( *""* /) 
each term of which must be separately zero, consequently each term in our 
expansion of r n P n (p,) is a solution of Laplace's equation and is a poly- 
nomial of degree n in x, y and z. Similarly, if r', 0' , c/>' are regarded as 
polar co-ordinates of a point (x' 9 y r , z'), r' H P n (IJL) is a solution of Laplace's 
equation relative to the co-ordinates of this point. We infer then that 

F n (0, 8') - A n m P n m (cos 0) P n - m (cos 0'), 
where A n m is a constant to be determined. 
We thus have the result that 



S n (0,<j>)= S n m P m (cos 6) e"*, 

m=*n 

where B n - . ^ n m f f ^ F (0 f , <f>') P n ~ m \cos 9') er*' sin 0'd9'dcf>'. 

477 J J 

To determine the constant A n m we consider the particular case when 

V = r n P M m (cos 9) e im *, 
F (9', </>') = a n P n m (cos 0') e*+\ 
then (2i/+l) S, (fl, <) - a n P w ^ (cos (9) e'* i/ - n 

= ^=^w, 

and consequently 

,2 4- 1 

X - -4^ 

or ^4 n w ( ) m . 

24-2 



A n m ^ f "" P n m (cos 0') P n - (cos 9') sin 0'd9'd<f>' 
Jo Jo 



372 Polar Co-ordinates 

Hence we have the expansion 

P n (p) - S (-) m P n m (cos 6) P n ~ (cos 0') e"<*-*'>. 

m**n 

6*35. Legendre's expansion. Transforming the last equation with the 
aid of the relation of Rodrigues, 

U ~~ m 



-m /..\ __ 
- 



we obtain Legendre's expansion* 

00 (rvi _ . Yf) \ t 

P n (p) = 2 S - - j P n m (cos 6) P n (cos 0') cos m (< - f ) 

ml \'^ ' '"7 i 

+ P n (cos 0) P n (cos 0'), 
and the expression for B n m may be written in the alternative form 

1 (rt 



f f w 

1 1 



One simple deduction from the expansion for S n (6, <f>) is that a simple 
expression can be obtained for the mean value of S n (9, </>) round a circle 
on the sphere. Let the circle in fact be = a, then the mean value in 
question is obtained by integrating our series for S n (9, </>) between 0=0 
and $ = 2-n- and afterwards dividing by 2ir. The result is that 



tin (0, </>) = B n P n (cos a). 

Now when 0=0, P n m (cos 9) = except when m = 0, and then the 
value is unity, hence 

S n (0, 0) = ,B n o, 



and so S n (6, <f>) = flf n (0, 0) P n (cos ), 

where the coefficient S n (0, </>) is the value of S n (9, <f>) at the pole of the 
circle. This theorem may be extended so as to give the mean value of a 
function / (9, </>), which can be expanded in a series of type 



The result is ni=0 



n=0 

If the analytical form of the function /is not given, but various graphs 
are available, the present result may sometimes be used to find the 
coefficients in the expansion 

/(0,<)= 2 C' n P n (cos0)+ S E P w m (cosfl)[-4 n m cosm^ + 5 n m sinm^]. 

n=0 n==l m=0 

To use the method in practice it is convenient to have a series of curves 
in which / is plotted against <f> for different values of and a series of 
curves in which / is plotted against 9 for different values of <f>. The two 
* Legendre, Hist. Acad. Sci. Paris, t. n, p. 432 (1789). 



Expansion of a Polynomial 373 

meridians <f> = /3 and <f> = /? 4- TT may be regarded as one great circle with 



Mean values round "parallels of latitude" for which 6 has various 
constant values will give linear equations involving only the coefficients C n . 

Since P n (0) = when n is odd and P n m (0) = when n is even and 
m is odd, the mean values round meridian circles will give equations 
involving only the coefficients A n m and B n m in which both m and n are 
even, but terms of type C n will also occur. To illustrate the method we 
shall suppose that the function / (9, <f>) is of such a nature that spherical 
harmonics of odd order or degree do not occur in the expansion and that 
a good approximation to the function may be obtained by taking terms 
of orders and degrees up to n = 4 and m 4. We have then to determine 
the nine coefficients <7 , C 2 , 64, A 2 2 , 2 2 , ^4 4 4 , J5 4 4 , ^4 4 2 , J3 4 2 . Three of these 
may be determined from the mean values of / round parallels of latitude, 

say = ^, = =, = -. Two equations connecting A 2 2 , A^, A 4 2 may be 
2 o u 

obtained from the mean values of / round the meridians </> = 0, TT and 

<f> = ~, - - , while two equations involving jB 2 2 , JS 4 2 , A may be obtained 

2i . 

from the mean values round the circles <A = ~ , -.- : J> = r , c6 = - . 

^ 4 4 r 4 T 4 

Further equations may be obtained from the mean values round the circles 
,, 77 ITT , 77 4?r , 2?r STT , 5?r HTT 
^ = 6' T ;< ^ = 3' "3 ; * = "3"' '3"^^ 6 J ~6~- 

Having found CQ, (7 2 , C 4 from the first three equations and having 
expressed A 2 2 , A^, JS 2 2 , 5 4 2 in terms of Af with the aid of the next four, 
two of the last set of equations can be transformed into equations for 
Af and Bf. 

When the two sets of curves have been drawn the mean values of / 
round the different circles may be found with the aid of a planimeter. 

6*36. Expansion of a polynomial in a series of surface harmonics, 
When r n F (9, <f>) is a polynomial of the nth degree in x, y, z the expansion 
of F (0, $) in a series of surface harmonics may be obtained in an elementary 
wav by using the operator V 2 . Let us write r n F (9, <f>) = f n (x, y, z). The 
first step is to determine a polynomial / n _ 2 (x, y, z) such that 

fn (X> y, Z) - ?* 2 /n_.2 (X, y, Z) 

is a solution of Laplace's equation of type r n S n (9, <f>). The equation 

V 2 [/ n -r*/ n _ 2 ]=0 

gives just enough equations to determine the coefficients in/ n _ 2 . To show 
that the determinant of this system of linear equations does not vanish 
we must show that y 2 r r zf ] =t o 



374 Polar Co-ordinates 

If, however, r 2 / n _ 2 were a spherical harmonic of degree n we should have 
(^ 2 / n _ 2 )/ w -2^ when integrated over the spherical surface, because / n _ 2 

can be expressed in terms of surface harmonics 8 m (9, </>) of degree less 
than n. But this equation is impossible unless / n _ 2 vanishes identically. 

Having found / n _ 2 we repeat the process with / n _ 2 in place of f n and 
so on. We thus obtain a series of equations 

f __ r 2f __ r 2.Qf 

J n ' J n2 ' ^n J 

f T 2 f = T 2 S 

from which we find that 

When n = 2m, where w is an integer and/ n = /, the spherical harmonics 
are determined by the system f)f equations* 

V 2 "-/- (2m, 2) (2m 4- 1,3)S , 

V 2w - 2 / = (2m, 4) (2m 4- 1, 5) r 2 # -I- (2m - 2, 2) (2m 4- 3, 7) r 2 S 2 , 
V2m- 4 y = ( 2m , 6) (2m 4-1,7) r 4 /S 4- (2m - 2, 4) (2m -f 3, 9) r*8 2 

4- (2m - 4, 2) (2m 4 5, 11) r 4 /S Y 4 , 

where (a, 6) = a (a 2) (a 4) ... 6. 

Solving these linear equations we find thatf 
r'^Sw (2m - 2k, 2) (2m 4- 2k 4- 1, 4fc 4- 3) 

r 4 

+ 2.4~(4]fc -" 

where 2fc (r, y, 2) = V 2m - 2k f (x, y, z). 

The equivalence of the two expressions for S is a consequence of 
Hobson's theorem ( 6-22). 

There is a corresponding theorem for a space of n dimensions. The 
fundamental formula for the effect of the operator 

is V w 2 (r 2 %) - 2p (2p -f- 2q 4- n - 2) r 2 *>~% -f r*V n *v Q , 

where r 2 = o:^ 4- x 2 2 4- ... x n 2 , t; v = V Q (x l9 x 2 , ... x n ). 

* If VQ (x, y, z) is a rational integral homogeneous function of degree m, we have 

V 2 (r*%) = 2p (2p + 2q + 1) r 2 ?- 2 ^ -f r 2 P V 2 v . 

Hence if V 2 v Q = the effect of successive operations with V 2 is easily determined. 
f G. Prasad, Math. Ann. vol. LXXII, p. 435 (1912). 



Legendre Functions of i cot 6 375 

The equations are now 
V n **/= (2m, 2) (2m + n - 2, n) S , 
V n 2 - 2 / - (2m, 4) (2m -f 7i - 2, 7i -f 2) r 2 S 4- (2m - 2, 2) (2m + n, n -f 4) r 2 S 2 , 

r 2 *^ (2m - 2i, 2) (2m -f- 2fc + n - 2, 4fc + n) 

r2\7 2 r 4 V 4 1 

* 4_ _ _ n V2m-2fc/' 

2(4fc + n- 4) ' 2.4(4fc + ra- 4) (4i + rc- 6) '"J ' 

__ r 4 *+*- 2 _ / a a a\ 2 _ n 

- (ifc - 4 + w, n - 2) fe Va^' 8^ 2 ' '" dxj T ' 
where 2fc (^ , o: 2 , . . . x n ) - V n 2w| - 2 */ (^ , a? 2 , . . . a: n ), 

and / is a homogeneous polynomial of degree 2m. 

6-41. Legendre functions and associated functions. It should be 
observed that Laplace's equation possesses solutions of type 

r n P n m (/A) e* tm *, rQ n m (p.) e tm *, 

when n and m are any numbers. It is useful, therefore, to have definitions 
of the functions P n m (/x) and Q n m (/z) which will be applicable in such cases 
and also when fi is not restricted to the real interval 1 < /* < 1 . 

The need for such definitions will appear later, but one reason why 
they are needed may be mentioned here. 

In an attempt to generalise the method of inversion for transforming 
solutions of Laplace's equation* it was found that if 

Y - Jl^-? 2 7 - - t r * + a ~ Z = - *- (I) 

2(x-M/)' 2(x-iy)' x-iy' (> 

an(J if / (X, Y, Z) is a solution of Laplace's equation in the co-ordinates 
X, Y, Z, then (x-iy)-lf(X, Y,Z) 

is a solution of Laplace's equation in the variables x,y,z. Introducing 
polar co-ordinates, we find that 

R = iae 1 *, r = iae 1 *, sin = cosec 9. 

The standard simple solutions of Laplace's equation give rise, then, to 
new simple solutions of type 

(sin 0)~* P n m (i cot 9) c t(n+ i>* r ~l +m , 

and we are led to infer the existence of reciprocal relations between 
associated Legendre functions with real and imaginary arguments and of 
more general relations when the arguments are complex quantities or real 
quantities not restricted to the interval 1 < z < 1 . 

Definitions of the associated Legendre functions Q n m (z) for all values 

* Proc. London Math. Soc. (2), vol. vii, p. 70 (1908). 



376 Polar Co-ordinates 

of n, m and z have been given by E. W. Hobson* and by E. W. Barnesf. 
The definitions adopted by Barnes are as follows : 

Let z = x -h iy, w = log - , 

25 1 

2r(l-ra-.s) j mu , 

e 



then, if | arg (z 1) | < TT, 

^n (^) == sin 7&7T y (m, n, 5) (2 I) 8 d#, 

where the integral is taken along a path parallel to the imaginary axis with 
loops if necessary to ensure that positive sequences of poles of the integrand 
lie to the right of the contour, and negative sequences to the left. Also 

Q n m (z) = e ~ m 



where I m = TT cosec n7r.P n m (z), and the upper or lower sign is taken in the 
exponential factor multiplying I m according as y ^ 0. 

The functions P n m (z), Q n m (z) are not generally one-valued. To render 
their values unique a barrier is introduced from oo to 1. When'w is 
not a positive integer and z is not on the cross-cut, P n m (z) is expressible 
in the form 

P n m (^ - p -~_ - } A mw F{ - n, n + 1; 1 - m; J (1 - )}, 

where F (a, b ; c ; x) denotes the hypergeometric function or its analytical 
continuation. This formula, which gives a convergent series when 
| 1 z | < 2, shows that z = 1 is a singular point in the neighbourhood of 
which P n m (z) has the form 

(,_ 1)-** {(70 + ^(2- 1)+...}. 
Under like conditions 

2Q n m (z) . sin rnr. F ( m ri) 



* -f 1 

where, as before, e w = - - . 

z 1 

The definition of Q n m (z) given by Barnes differs from that given by 
Hobson, the relation between the two definitions being given by the 
formula 

sin HTT [Q n m (z)] B = e~ im " sin (n + m) 7r.[Q n m (z)] H . 

* Phil. Trans. A, vol. CLXXXVII, p. 443 (1896). 
4- n.**f I M .~* %i WVTV ^ OT /icmo\ 



Relations between the Functions 377 

It follows from the definitions that 

P B (- z) = e*-* P.- (z) - 2 5^T (2 ), 

#n m (-2) = -<?"> (Z).e"", 
P"-,-! (Z) = P n m (2), 
"__! (Z) = <? (2) - 7T COt Tfor.P," (Z), 

V m <*) ." 1 ( sin 



g- m (z) r (m - n) = & (z) r (- m - ). 

When m = 0, or when m is an integer, P n m (z) has no singularity at 
z=l. This is evident from the expression for P n m (z) in terms of the 
hypergeometric function in the cases when m is negative or zero and may 
be derived from the formula 

- rl ~ m + n ) 



n - n (i+ -_ 

in the case when m is positive. We add some theorems without proofs. 

1. The nature of the singularity of P n m (z) at z = 1 may be in- 
ferred from the formula 



+ e ~ m r ( - nmr+ n) F{1 ~ m + n ' ~ m ~ n > l ~ 

where = | (1 -f z). When m is a positive integer, 



x F{m n, m + n+ l;m+ 1; J (1 )} ....... (A) 

2. When in addition n is an integer there are three cases: 

(1) < n< m. In this case P n m (z) = but r (1 + n m) P n w (2) is a 
solution of the differential equation. 

(2) n> m. In this case the formula (A) is valid. 

(3) n < 0. In this case, if n > m, 

p m (z) _ fc2 _ Dim T (m - 71) __ 

^n W-iz i) 2T(-m-n)r(m+l) 

x ^{w n, ra-fw+ 1 ; m + 1; (1 z)}. 

3. If - n < m, P n m (2) = 0, but F (- m - n) P n m (z) is a solution of 
the differential equation. 



378 Polar Co-ordinates 

4. When m = and n is not an integer and is not zero, 



= s rji ~ n ll (?LJ _+J) 0< 

x {log - 20 (1 4- 4- ^ (* - ft) + ^ (ft + 1 -f OK 
where = i (1 4- 3) and (?/) = --.- log F (%) 

2 V / V \ / fa fe \ / 

Hence P n (z) has a logarithmic singularity at x 1, at which it 
becomes infinite like 

TT~ I sin UTT . F { n, n -f 1 ; 1 ; 0} log -f a power series in 0. 

5. When m 0, we have seen that P n (z) has a cross-cut from oo 
to 1 ; when, however, \m is not an integer and not zero, P n m (z) has a 
cross-cut from oo to 1, and is therefore not defined by the preceding 
formulae when 1 < 3 < 1. It is convenient to have a single value of the 
function in this interval, and one which is real when m and n are real. 
It is therefore assumed that as e -> and 1 < x < 1, 

P n m (x) -~ lim e* mnl P n m (x -f i) - lim e-^ mt p n m ( x _ i) 
1 /I -h x\ 



- 7/ ~~ ,- m; - 



n (x) - I lim {Q n >* (x + ei) + Q n (x - i)} 

= _ _ - ^ 1^.. . [0 Cm> (X ) + 

2 sin H,TT I ( m - n) 1 ' 



where 



6. The function Q n m (x) has a cross-cut between 1 and 1. For values 
of z for which | z \ > 1 the function can be expanded in a convergent power 
series in 1/3. If | arg (z 1) | < TT and (2- - l) irn - (z - l)* m (2 + l) im , 

/ ,n /x sin > + w) ^ r ( + m 4 1) T (J) ( 2 a - l) Jw 



x F(Jn-f Jm -f 1, jn-f |m + J;w + f ;- 2 ). 

The values for cases in which | 3 | < 1 may be deduced by analytical 
continuation of the hypergeometric function and use of the foregoing 
definition when z is real. 



Reciprocal Relations 379 

6*42. Reciprocal relations*. Barnes has shown that the power series 
in l/z can, under the foregoing conditions, be expressed in the form 



O (z\ ~ C 

Vn \Z) ~ O -- 



; T- , 

i z j 



sin (n -f m) TT T (n + m + 1) T (4) 
"" '"sin nit " 2" r (n -f |) ' 

Putting z = i cot (9, we have 
(> n m (i cot 0) - Ci- n - 1 sin** 1 

x ^"{| (n + m + 1), \ (n 7>i + l);n -f |; sin 2 0}. 
Now 
F {J (n -f m 4- 1), I (n - m 4- 1) ; n 4- f ; sin 2 9} 

= F {n -f m + 1, n - m + 1 ; n + jj ; sin 2 0} 
- (cos ifl)- 2 "- 1 .F [m +J,J-w;n-f|; sin 2 10]. 
Therefore 

Q n (i cot 0) - CT 11 - 1 2 " f * (sin 0)i (cot |0)~ W " J 

x JP [m + A, i - m ; n + | ; sin J0]. 
But 



3 - (cot |0) n 4 F {m -f , J M; n -f- f ; sin 2 0}. 
Therefore 

Q n m (i cot 0) = . ^7 (1 sin 0)* P" n -* (cos 0). 

sin n-n . 1 ( m n) * -m-t 

Writing m | in place of ft and n \ , in place of m, the formula 
becomes 

Q~"~\ (i COt 0) -^4~ Tx (1 sin 0)* P n W (COS 0). 

^_ m _i v / cos m7T Y (m -f- n -f 1) 

Again, Barnes has shown that when | 1 z 2 | > 1, 



-f C 2 (2* - l) n ^ i (m - 7i), (- m - n); - n; 

I 1 

where 

2(--) 2Tj(n+ |) . 



* Judging from a conversation with Dr Barnes in 1908 he had at that time noted at least one 
explicit reciprocal relation between the functions P n m (z) and Q n m (z). 



380 Polar Co-ordinates 

consequently, using again the transformations of the hypergeometric 
series, we find that 



P (i cot 6) = (2* cosec 0)~ 

x .F{-hra, m; n -f- f ; sin 2 0} 

+ rlrT^ 1 !) 6l "" (cot ^ )n+ * -** tt~ ^J + ^;i-^;sin 2 ^} 

= - sin (w + I) TT. r (w + w + 1) (27r cosec 0)~* lim Ql*l\ (cos - fe), 

7T e->0 

and so 



P" n "* (i cot 0) = - - sin UTT . T (- m - n) (2n cosec 0)-* lim Q n (cos - t) 

~~ w "~* 77 e^Q 

= - ,- ------ r v--'/ --- - x - (27r cosec 0)~* lim ^ n m (cos - fc). 

r(l 4- m -f n) sm(m + n)ir *->o 

This is very similar to the reciprocal formula obtained by F. J. W. 
Whipple*, which may be written in the form 

n) ^ 



n m (cos h a) - ___m n ^ 
" v ; * - m ~* 



EXAMPLES 

1. Prove that when n is a positive integer 

{(/t2 - 1)n log * (1 + " )} - Pn 

[A. E. JoUiffe, Mess, of Math. vol. XLIX, p. 125 (1919).] 

2. Prove that if 2x = <i + t~i, 



i i, 1, 1 - I) 

= t* (2 log 2 - i log t) f (|, i; 1; 



I)' 



Prove also that 



- 1 [(4 log 2 - 4 - log ^ (?, i; 1; - 4 {(f) 2 (J - J) * 

+ i [(f ) 2 ~ (S) 2 ] (J + * - i - i) t* + ...}]. 
[H. V. Lowry, Phil. Mag. (7), vol. n, p. 1184 (1926).] 

* Proc. London Math. Soc. (2), vol. xvi, p. 301 (1917). 



Potentials of Degree n + \ 381 

3. If K is the quarter period of elliptic functions with modulus k and complementary 
modulus k* = (1 - & a )i, prove that 



V " 

K = ^ 



W 



,P- 



2p*"ni-jfc 



(1*-*) P (** i; * ; (T+l?) 



The last series is recommended by Lowry for the calculation of jfiT when k is nearly 1. 

4. Prove that if n is a positive integer the equation P n -f (z) = has no root which lies 
in the range 1 < z < 3. 

5. Prove that if 2n + 1 4= 



6. Prove that if n is a positive integer 

P n (l-2sin 2 asin*0) - [P n (cos 0)]* + 22 ~, [P n m (cos W cos 2m 
and deduce that f or < 6 < -n 



Hence show that the roots of the equation P n (cos 0) = can only lie within certain 
intervals in which sin (2n +1) is negative. 

6*43. Potential functions of degree n + J wAere n is an integer. If we 
apply the imaginary transformation (I) to the potential functions of 
degree zero and 1 we find that if /> 2 x 2 + y 2 and F ( ), O ( ) are 
arbitrary functions of their arguments, the functions 

F= (x- %)-**>- fe) + 



F = (3 + iy)-* ( - ) + Q 



are solutions of Laplace's equation. In particular, 

(x - iy)~* (P - *8) w+1 

is a homogeneous solution of degree n -f . Differentiating this A times 
with respect to x we obtain a solution which may be written in the form 



382 Polar Co-ordinates , 

The typical term of this series is a constant multiple of 



and when the derivative in this expression is expanded in powers of x, the 
coefficient of x k ~ s is 



Now x = -p [e*+ + 



consequently the term involving #*-* gives rise to a term in our series with 
the exponential factor e l(k+ fr* and a number of terms with exponential 
factors of lower order. Taking all the terms with the exponential factor 
e u+J) wc mus t get a solution of Laplace's equation and this solution is 
represented by the series 



v = c . <*+}>* 2 C>*-a-i ~ (p 

op, 



where 

1 ' W* s (SI)* (/C S) I 

We may conclude that if /(a) is an arbitrary function with a suitable 
number of continuous derivatives, the series 

'i a 

iz) 



is a solution of Laplace's equation. 

6-44. Conical harmonics*. When V is independent of </> a set of 
solutions suitable for the treatment of problems relating to a cone may be 
obtained by writing r = c,e 8 , V = (cr)~i w. The equation for u is then 

a du 



and there are solutions of type u = cos (ks) K (Jc) (p,), where K (k) (p,) is a 
solution of the differential equation 



Mehler writes 

2 r oo cos 

"- 



2 r oo cos a . 

(p) - - cosh (for) r 7"-T 
vr/ TT v ; Jo [2 (cosh a 



and remarks that 7i (fc) (/i), ^ (p (- p,) are two essentially different solutions 

* F. G. Mehler, JfcrfA. Ann. Bd. xvm, S. 161 (1881); C. Neumann, ibid. S. 195; E. Heine, 
Kugelfunktionen, Bd. n, S. 217-250, 



Conical Harmonics 383 

of the differential equation. The function K (k} (/x) can be expanded in a 
power series 

K (p.) = F j + W, J - H; 1 ; - 



_ 4P -f I 2 /I - /A , (^1+ I 2 ) (4 2 f 3 2 ) /I - /A 2 
-i-l- - 2a - -\^ 2~; + 22.42 V 2" /'"' 



and its relation to the Legendre function becomes clear. 
There is also an expansion 



where L - I + ** + A' ,. 

4fc + 3= 
^^^+ - 4 o M | - 4 .0.8.(10) 

Problems relating to a cone have also been treated with the aid of the 
ordinary Legendre functions*. 

If, for instance, there is a charge q on the axis of z at a distance a from 
the origin, and the cone = a is at zero potential, the potential V is given 
by the series 



sin' a ( ~? "p 1 ) 
V dn dp /_ 



r<a 



Sill 2 (- "- -" 

V 3n 3^ y^ a 



dp 

a, 



where the summations extend over all the positive values of n which 
make P n (cos a) = 0. 

EXAMPLES 

1. Prove that when < < JTT 

*<*> (cos 0) - J ( J + J , i - ; 1 ; sin^ B) 

= i + M 4 |.I 2 8in2 a + < V + n W + 

When JTT < 6 < TT the series represents K (k) (- cos ^). 

2. Prove that 



3. Prove that 

(r 2 + c 2 2rc cos 0)~& (rc)~i / . yj . K (k} ( cos 6) dk. 

J o cosh (7r) 

* See, for instance, H. M. Macdonald, Camb. Phil. Trans, vol xvm, p. 292 (1899). 



384 Polar Co-ordinates 

4. Prove that when k is large and positive 

K ( V (cos 0) ~ e w (l + S) (27r& sin 0)-*, 

where 8 - as & ~> >. The point = TT must be excluded, for IT (cos 0) has no meaning 
for = TT. 

5. Prove that 

(Jfc) cosh (for) / s-ifc. 

* (l " " --' 



7T J i V fl 

/2_ 
K (k) (cosh u cosh v - sinh u sinh v cos <) d< = 2<nK (cosh w) #<*> (cosh v). 
o 



6. If /(*)=/ # (fc) (*)tanh(fo7 

show that under suitable conditions 



The results of examples 1-6 are 'all due to Mehler. 

6-51. Solutions of the wave equation. These may be found with the 
aid of the Green's substitution 

x = s sin a sin /? cos 0, y = s sin a sin /? sin <, 

z = s sin a cos /S, i"c = s cos a, 
-f dy 2 + ^ 2 ~ c W = d^ 2 + 5 2 d 2 + s 2 sin 2 a d 2 + s 2 sin 2 a sin 2 /? . d^ 2 , 

d (a;, y, 2;, ^) = s 2 sin 2 a sin ]8 d (s, a, j3, ^>). 
The wave equation now becomes 



d*u 3 du 2 du 1 

" + + cot a + 



du 



^ s 2 sin 2 a sin 2 j8 3)8 V ^ 3j8; ^ 5 2 sin 2 a 
and possesses elementary solutions of the form 

u = s n A (a) B (]8) cos m (< </> ) 



_ 

~ ' 



^ + 2 cot a ^ + n (n + 2) - -^-f - 
da 2 da . [_ ' sm 2 a 

We may thus write 

B = CiP m * (cos j3) + c 2 Q m " (cos ]3), 
^ = Vcosec a [&i-P^+* (cos a) -f 6 2 $w+\ (cos a)], 

where c 1? c 2 ; 6 1? 6 2 are arbitrary constants. On account of the reciprocal 
relation between the Legendre functions we may also write 

A = cosec a [o^P"^2\ (* cot a) 4- ^Ql^Ii (i cot a)], 
where a t and a% are new arbitrary constants. 



Green's Substitution 385 

The analysis is easily extended to Laplace's equation in n -f 2 variables. 
The appropriate substitution is 

x l = r cos i9 x 2 = r sin 2 cos 2 , # 3 = r sin 9 l sin 2 cos 3 , 
x n+1 = r sin 0i sin 2 ... sin n cos 0, # n + 2 = r sin X sin 2 ... sin n sin <, 
and Laplace's equation becomes 



Wr L , ^ sin" 9, rin-i 8 Z ... sin n g -J + ^ ^-1 sin" 0, sin"- 2 ... sin 8 n ^ 

9 f-f. -/!-,/) -n^ 

9 ... sm0 M x, 



+ ;TJL f^' 1 sinn " 2 fl i sinn ~ 3 02 - cosec fl n ^1 = 0. 
ocf> [_ 9^ J 

Assuming that there is an elementary solution of type 

u = R (r) &i (0,) 2 (0 2 ) ... n ( n ) cos (m0 + c), 
we obtain the equations 

r/ 2 ) r7(^> 

+ cot n a ^ -f [v n K + 1) - m^ cosec* n ] n = 0, 

' + (n- ^ + 1) cot 8 ^- 5 + [ V8 (v a + n - * + 1) 

^a+i (^8+1 + n s) cosec 2 8 ] Q 9 = 0, 
?i 4-1 dR vMi) 



and so we may write 

n = A n P^ (cos ff w ) + J5 n , n m (cos n ), 
Q 3 - (cosec s )i<"-> [^ 8 Pu,/* (cos e a ) + 



where w s = v s + \(n s) y a 8 == v 8+l + $ (n s). 

If in place of Laplace's equation we consider the equation* 



the analysis is the same as before except that now the equation for R is 



d*R . n 
dr 

and the solution is 



dr> + r dr + r 2 "" r * ~ 0> 



If Vl = 5 ^n, where s is an integer, one of these Bessel functions must 
be replaced by Y 8 (kr), the second solution of Bessel's equation. 

* E. W. Hobson, Proc. London Math. Soc. vol. xxv, p. 49 (1894). 

25 * 



386 Polar Co-ordinates 

It should be remarked that the elementary solid angle in the generalised 
space for which x^ , # 2 , ... x n+2 are rectangular co-ordinates is 
dc* - sin" ^.sin"- 1 2 ... sin O n d (0 l9 2 , ... n .(f>). 
When this is integrated over all angular space the result is 



where s = n -f 2. See, for instance, P. H. Schoute, Mehrdimensionale 
Geometric, Bd. n, S. 289. 

For ordinary space ^ =- 1, and the foregoing result tells us that the 
equation V 2 V f k 2 V ----- is satisfied by 

V = r~l ,/,+j (kr) P s m (cos 0) cos w<. 
In particular, when s = we have the solution 



and when s I there is a corresponding solution 

V = r-U_j (Ar) = (2/ftTr)* ~ 8 . 
These may be combined so as to give the solution 

eikr 

V = r ' 

which arises naturally from the wave-function 

y _ e ifc(c<-r) 

r 

suitable for the representation of waves diverging in three-dimensional 
space. This type of wave-function is fundamental in the theory of Hertzian 
waves and also in the theory of sound. The general function r~^J K ^(kr) 
can be expressed in the form 



and is easily seen to be of the form 

P 8 (a) sin a + Q s (o) cos a, 
where P and Q are polynomials in a" 1 . 

For some purposes it is convenient to use the notation 
^(x)=(|77a,-) i / n+i (x), 

Xn (^) = (-)" (***)* J-n-\ (X) = - (1**)* F n+i (X), 
Tin (X) = (*) - iXn (X), 
Cn (*) = -/ (*) + ^n (*) 



Representation of Diverging Waves 387 

These new functions 7? n , n are connected with Hankel's cylindrical 
functions by the relations 

r, n (x) = (^)i JV+* (x), (x) = (^)i # 2 "+* (*). 
When | a; | is large and | arg x \ < TT we have the asymptotic expansion* 



r) n (X) = (- t)e" 



The series terminates when n is an integer and gives an exact repre- 
sentation of the function. In this case we may write 



and when kr is real a series for | n (AT) | 2 may be obtained from the linear 
differential equation of the third order satisfied by [i/j n (kr)] 2 , [% n (kr)] 2 and 
ifj n (kr) Xn (kr). This series 

| (kr) | = 1 + in (n + 1) ( -^- )a + * ;* (n -\)n(n+l)(n+ 2) ^L 

1.3... (2/i- 1)., 1 
f -- 2.4!.. 2n (2tt! W' 

which contains only positive terms, shows that | n (ir) | decreases as kr 
increases, hence, if a series of the form 

2 t n (kr)f n (0,<f>) 

n-O 

converges absolutely for any value of kr greater than zero, it converges 
absolutely for all greater values of kr. 

For small values of | r \ the function ifj n (kr) is represented by the series 

_(* r ) n+i _r. (^ , _(^ __ i 

1.3... (2n+ 1) L 2(2n+ 3)" t "2.4(2n f 3) (2n- + 5) "J ' 
which converges for all values of r. For large values of | r | we have 
approximately 

<An (kr) = i [i^i-ic-'^ + (- i) n+1 e tfcr ], 

This formula is exact when n = and i/j n (kr) = sin AT, and when 
n = 1 and ^ n (kr) = cos AT. In other cases it represents simply the first 
term of an expansion which terminates when n is an integer. It should be 
remarked that 

. sin AT 

0i (^) = r - cos AT, 

[31 3 

77 ,- -- 1 sin AT r- cos AT. 
(A^r) 2 J kr 

* Whittaker and Watson, Modern Analysis, p. 368. 

25-2 



388 Polar Co-ordinates 

The functions ^ n (x) may be calculated successively with the aid of the 
difference equation 

(2n + 1) n (x) = x [^ (x) +> n+1 (*)], 
while their derivatives may be calculated with the aid of the relation 



(2n + 1) - = (+!) ^ (*) - 



(x). 



Similar relations may be used to calculate the functions and their 
derivatives. 

These functions are particularly useful for the solution of problems 
relating to the diffraction of waves by a sphere*. 



1. Prove that if z = r\i 



and deduce that 



EXAMPLES 



tBauer 



2. Show by means of the result of Example 1, that 
I" -W^-W^- 

J CO ** 



3. Prove that 



sin fc r 2 -4- a 2 



4. If K 2 = r 2 -f a 2 - 2ra/x and n = e~ iks /Il, prove that 



[Clebsch.] 



where f n (kr) = t, n (ka) $ n (kr) or n (kr) $ n (ka) according as r is less than or greater than a. 

[Macdonald.] 

5. Prove that $ n (ka) n ' (ka) - { n (ka) n ' (ka) - i. 

6. Prove that if R 2 = r 2 + a 2 - 2ar/i 

sin (&J?) sin (Ar) sin (ka) 



l" 1 cos(^) sin (fer) cos (Jba) 

"^"^- 



cos (Ar) sin 



r<a 



- - 
ka 



[Eayleigh.] 

* See for instance, Lamb's Hydrodynamics, 5th ed. p. 495; H. M. Macdonald, Proc. Roy. Soc. A, 
vol. xc, p. 50 (1914); G. N. Watson, ifetd. vol. xcv, p. 83 (1918); Lord Bayleigh, Phil. Trans. A, 
vol. ccm, p. 87 (1904); Papers, vol. v, p. 149; A. E. H. Love, Proc. London Math. Soc. vol. xxx, 
p. 308 (1899). 



The Method of Stieltjes 389 

6-52. There is a second method of 'dealing with the homogeneous 
wave-functions which is due to Stieltjes*. 
If we write 

x = u cos <f>, y u sin <, z = v cos x, w = v sin x> 
the equation D W = becomes 

18JF 1 8 2 TF 8 2 tiT 



This equation possesses solutions of type 

W = u m v k e i( *+ k * } f (u, v) 



This equation belongs to the class of "harmonic equations" studied by 
Euler and Poisson, it retains the harmonic form in which the variables are 
separated when the variables u and v are subjected to a number of 

transformations of type . ^ , .. 

JF w + tt; = JF ( + *T)). 

The general theory of these transformations has been discussed in 
Ch. IV. At present we are interested only in the particular substitution 

u + iv = se iB , 

which transforms the equation into 
3dW 



_ __ _ _ 

s ds s* 80 2 s* cos 2 8< 2 s* sin 2 6 



Putting cos 20 = ^ we find that there are elementary solutions of 
form 



w = 

where satisfies the differential equation 

2 1ft, \ 

n A_. .1 = 

+ /*) 2(1- ,*)[ 

(B) 

This equation is satisfied by 

m k 

where 2jp = m + k and F as usual denotes the hypergeometric series. If 
n p is a positive integer the series terminates, and if n is also a positive 
integer we obtain a solution in the form of a polynomial. 

* Comptes Rendus, t. xov, p. 901 (1882). 



390 Polar Co-ordinates 

Writing v = n p we have a solution of equation (A) which may be 
written in the equivalent forms 

f=(u*+ v 2 ) v F (v + m + k + 1, - v\ k + 1 ; r) 



- v\m+ 1; 1 T) 

/ v 2 \ 

- u 2v F ( - v, - m - v\ k + 1 ; --- J 

M / 



u ?; i 

where r = r - , 1 r = . 

M 2 + t; 2 ' M 2 + v* 

To express a given wave-function in a series of elementary wave- 
functions of the present type we need an expansion theorem relating to 
series of hypergeometric functions. A formula for the coefficients in such 
a series was given long ago by Jacob! and is derived in 6-53. The con- 
ditions under which the series represents the function were investigated 
by Darboux and have been studied more recently by other writers. 
Corresponding studies have been made of other series of hypergeometric 
functions. Some references to the literature are given in Note III, Appendix. 

An interesting reciprocal relation between solutions may be obtained 
by making use of the fact that if 

Y * 2 ~ a2 V ' s * ^ aZ 7 - a ~ W aw 
2 (x iy) ' 2 (x iy} ' x iy ' x iy ' 

and if / (.Y, Y, Z, W) is a solution of 

2 

~ ' 



then v = (x - iy)-*f(X, 7, Z, W) 

is a solution of S 2 v S 2 v S 2 v d 2 v _ 

dx* + ty* + a7 2 + dw* = ' 

when considered as a function of x, y, z, w. Now, if 
x ~ s cos #cos <f), y s cos sin </>, ^^^sin^cos^, ^ = 5 sin sin ^, 
X=/Scos0cos0, 7=/8fcos0sinO, Z = /Ssin0cosX, TF-5 sin sinX, 
we have the relations 

S 2 = - aV^, 5 2 - - 2 e 2i<1> , cos - sec 5, X = x> 
and so a solution 

S n cos w sin fc 0c z(m * ffcx) G (n, ?/i, jfe, cos 0) 
corresponds to a solution of type 

- (S cos6e-^)- l .Bec m (i tan 9) k .(s/ia) m .e ik * G (n, m, fc, sec 9) (ia) n e n< * 
i.e. s" 1 - 1 (cos 0)-i-*-* (sin 0) fc e (w+1)I * + *^ (w, m, ^, sec 0). 



Reciprocal Relation 391 

EXAMPLES 
1. Prove that if 
(w, m, k) = s n cos m sin* B.e'^+W F (%n + 1 + w + i^^rn + fc - in; & + 1; sin 2 0) 



__ 
then r(*4-l)r(in-im-p+l)' 

> m ' *0 = 2 (n - 1, m + 1, jfc), 



(-r i ~ - J (n, m, A:) = | (n + m k) (n + m 4 &) (n 1, m 1, k), 

(a + * b~ I ( n m > ^) == (& n w) (w 1, w, & + 1), 
Xc/^ c/i^/ 

(~ i ~ } (n, m, k) = (n + m -\- k) (n 1, m, k 1). 

\(7z a^/ 

2. Prove that if w = y 2, v = z x, u' =- x y, the differential equation 



. _ rl 

W _ I 



possesses a solution of type 

^ - y - m 2 +y+ a -2ft - 2m 

dj 

When this solution is a homogeneous polynomial of degree m in x, y and z it can be 
expressed in six different ways in terms of the hypergeometric function, the arguments 
being respectively 

V W W U U V 

u 9 u 9 v 9 v 9 w 9 w' 

3. If u + iv = a cosh (a -f if$) 

and x u cos <, y u sin <j>, z v cos #> ict v sin x, 

the wave-equation becomes 

dW dW 



-5 * +" -^^ + 2 coth 2 a - -f 2 cot 2)3 -^ _ + (cosech 2 a -f cosec 2 j5) -^-5- 
aa- op- da. oft ox 

+ (sech 2 a sec 2 j9) -^-^ = 0. 
^ 

Hence show that there are simple solutions of type 

W - Ae (cosh 2a) 8 (cos 2)5) e* 
where A*, m and 4 are arbitrary constants and B (/^) satisfies the differential equation (B). 
4. Prove that the differential equation 

PV v &v ia*F 

a? + a?/ 2 + 'W " c 2 a^ + 7 * K ~ 

possesses simple solutions of the types 

V = S- 1 J 2n+1 (^) e (/it) e**+**x, 

F = J w (Ati cos a) J fc (Av sin a) e tw "H- ft x, 
and deduce the expansion of the second solution in a series of solutions of the first type. 



392 Polar Co-ordinates 

6-53. Jacobi's polynomial. The function 

H n (r) = F (n -f m + k + 1, - n\ k + 1 ; r) 
may be called Jacobi's polynomial*. It may be expressed in the form 

(k + 1) (fc + 2) ... (k + n) T* (1 - T) H n (T) = ~ [r*+ (1 - T)+] . 

...... (C) 

With the aid of this formula and integration by parts it is easily seen 
that 



r (n+ i) __ [r (fc + i)] 2 r (m + wjM_)_ 

+ i + ~2rT+ 1 r~(ra ~Mi + & +1) T (F+~w + 1)' 
A: > 1, m > 1, 7i / 



When m -f- k = 2p, m k = 2q where ^p and g' are positive integers, the 
polynomial can be expressed in terms of the Legendre polynomial with the 
aid of the formula 



p>q. 

Many interesting expansions may be obtained by expanding special 
solutions of the equation (A) in series of Jacobi polynomials. A few of 
these will now be mentioned. 

In the first place we have the two associated expansions 

(i - f - -nYH, (jr^-, i) = 2 A n H n (f) H n (,), 

\s + t) iy n _ 

H. (f) H, (,) = f fi, (1 - | - r,)' H 

p~Q 

where 

4 n = (m + k+2n+ 1) 

r(7i + A;+ l)T(p+ 1) T (p + m 4- 1) T (n + m ~f k + 1) 






m -f 1) T (p - n+ 1) T (n -I- 1) T (k -f 1) r7p + n + m 
+ 

r (g + i) r (m + ^ + s + p + i) r ($ + m + 1) r (A? + i) 



A proof of these relations may be based upon the fact that if we make 
the transformation 

*=(!-) (1-7,), *=-,, 

* C. G. J. Jaoobi, Crete's Journa 1 ,, Bd. LVI, S. 156 (1859); Werke, Bd. vi, S. 191. 



Various Expansions 393 

and take and 77 as new independent variables the equation becomes 



= 77 (1 - T?) 2 - [(m + 1) 7, + (4 + 1) (i, - 1)] , 

and is consequently satisfied by / = # 8 () # s (77). 

To determine the coefficients B P in the expansion of this solution in 
terms of the solutions already found it is sufficient to put rj = and to use 
the expansion 3 

H s ()~ 2 B p (l-f)*, 

p=0 

which is already known. To find the coefficients A n we put 77 = in the 
other expansion; we have then to find the coefficients in the expansion 

(1 - ) = I A n H n (). 

n-0 



This may be done by evaluating the integral 

i: _ 

: ['(I- Y f t ~ 
)Jo * 



Jo 

r (k + 1) 



_ f\m+n 



- Jli i ji- 1 l r i^ + - 1) _ f 1 

1 (k -i- n + 1) T (p - V-f 1) J 
rj& +JL) T (p + JL]_Tj(w + p + 1) 



A ^/t-r AJJ. v/ 7 ' ^/^ V'' 1 ' ' 1? ' x / 
1=3 r(pIT"nT 1 ) T (m^+TTw" -f ^> + 2) ' 
In the particular case when m = k = we obtain an expansion which 

ir V* ixmi'f.'f.on in f.Vi/a < frkT*m 



may be written in the form 



When /x,' = 1 this gives the well-known expansion 

,g 

( * + } 



The second expansion gives 

(/*) P, M = JS o (- r [T ^ r - j"^^; ^+7) ( -^ ) P 
If we write 



394 Polar Co-ordinates 

the 'quantities /x and // are the roots of the equation 

2 - 29x 4 2xy - 1 = 0. 
In particular, if y = 1, we have // = 1, /x ^ 2x 1. 

6-54. Some further interesting results may be deduced from the fact 
that if V (x, ?/, z) is a solution of Laplace's equation, then 

W (x 4 iy)~^ V (Vx 2 4 # 2 > z, ic) 
is a solution of the equation. 

In particular, if we take the solid harmonic 

V = r' n P n m (cos 6') e 1 *', 

m 
in / i i i \ 

where P n w (//) - \\ i/ i n ( ! ~ ^ 2 ) 2 C^ w (M), 

Z 1 (?// , i j 1 (n til -h J ; 

C n m (fji) -- ^ fm i w 4 1, m - n; w 4 1 ; l "- ^) , 

V ^ / 

we obtain the wave-function 

(^ 2 4 v 2 ) n e <w x-J p 

V ' " \ / ., 

'^t/, 2 4 ^ 2 

Comparing this with the type of wave-function already obtained we 
find that 

,1 i ^ n ( m 4 n 4 1 m n ^ , \ 

^n w (^) ^ ^ o ' o ; w 4 1 ; 1 u 2 n m even, 

\ L L ] 

fm -{- n \ 2 m- n + I \ 

^- /uJM , 9 ; m 4 1 ; 1 /x 2 1 n m odd, 

\ *j ^ / ^ 

the conditions under which the series terminates being given on the right. 
The expression (C) for Jacobi's polynomial now gives the interesting 
formulae* 



where 

Writing 

f 1 O 100! 

f = 1 - /i-, V) - 1 - i;-, a 2 = 1 



--- 

-f 17 1 



* Given explicitly by A. Wangerin, Jahresbericht deutsch. Math. Vercin, Bel. xxn, S. 385 (1914). 
The formulae are both included in the general formula given on p. 122 of the author's paper, 
Proc. London Math. Soc. (2), vol. m, p. Ill (1905). 



Products of Legendre Functions 395 

the two expansion theorems give 



- 2p F( _ 

(/*' + " 2 - l) p+ * 



EXAMPLES 

1. Prove that the differential equation 

d 2 y Vrn (m 1 ) n (n 1 ) 7 9 ~] 

j * = y\ -2 H - 2 - fc 

ax 2 [_ sm 2 x cos 2 x J 

is satisfied by 

y sin x cos n re jP ( , -= ; m + J; sin 2 a; J . 

[G. Darboux, Theorie dea Surfaces, t. n, p. 199 (1889).] 
Show also that 

dx 2 [_ cosh 2 x J 

is satisfied by y = e kx F - n, n -f 1 ; 1 - k; ~ ^ n ^ x \. 

2. Prove that 



r (m H- I)] 2 r (n + tfi + 1) (2a + 1) P 8 (M) n 4. I < 2 

- -' I 1 " 1 "'*' 



n (cos ) P B (sin e) = (^Y^ [cos- , sin" , - 2 - C $f _-$ 



--- _ 

2.4(2-l)(2n-'3)(ai"-6)(2-T) coa "'' 



3. If u = a;" 2 prove that 



1 d 



[L. Koschmieder, J?ev. JfcTa^. Hispano-Amer. (1924).] 

6-61. Definite integrals for the Legendre functions. Some useful definite 
integrals for the representation of Legendre functions may be obtained by 
deriving Newtonian potentials from four-dimensional potentials by inte- 
gration with respect to one parameter. 



396 Polar Co-ordinates 

Starting with the four-dimensional potential 
W'=f(x + iy,z + iw), 
we derive a second potential W from it by inversion. 

If s 2 = a; 2 + t/ 2 -f z 2 -f w 2 we have 



and a Newtonian potential is derived from this by integrating with respect 
to w either between oo and oo or round a closed contour in the complex 
w-plane. In particular, we may obtain in this way a Newtonian potential 

1 f (x+ iy) m (z -f iw) n ~ m dw 
"^J-oo (x* + y 2 + z 2 + w*) n + l ' 

which may be expected to be a constant multiple of r- fl ~ 1 P n m (cos 0) 
We thus obtain the formula 



which is certainly valid when n and m are positive integers as a simple 
expansion shows. The corresponding formula for P n (jz) is 



and the formula for P n m (p,) may be derived from this by differentiation. 

The last result may be obtained directly by expanding both sides of 
the equation 



*/ _i- z _ /7\ _ ______ _______ 

y + (z a) -^^ x * + y *+ (z _ a)2 + (w _ i 

in ascending powers of a and equating coefficients. The general formula 
may likewise be obtained for positive integral values of m and n by 
expanding the two sides of the equation 



W J-o 



dw 



- 6) 2 -f- (y- ) + (z - a) 2 + (ti>- ia) 2 
in ascending powers of a and b. We thus obtain the expansion 



00 



[(x - 6) + (y - i&) 2 + (z - a)]- =22 



n 0m0^! r n + 

which is easily obtained from the Taylor expansion 

71 If"^ 

P n (cos 6 -\- k sin 0) = 2 r P n m (cos < 

m O ^ 

by writing k = - e i<fr . 



Definite Integrals 397 

A formula for Q n (p,) due to Heine* may be derived from the fact that 
if t 2 = x 2 + y 2 + w 2 , the function 

W = t~*f(z+it) 
is a four-dimensional potential function. 

Writing w = p sinh u, where p 2 = x 2 + y 2 , the integral 

1 f 
V = -\ Wdw 



takes the form 

1 f 
V = ~ f(z + ip c sh u) du. 

7T J oo 

With suitable restrictions on the function / this integral represents a 
solution of Laplace's equation. 

In particular, if x, y and z are not all real quantities 

. (z\ , f 00 , . iv ! 7 
r- n - I (J n ( - 1 = I (z + ^p cosh u)~ n -*du 

f 00 1 

or Q n (a) = J [5 -f (s 2 - 1)* cosh u]~ n ~ l du. 

J CO 

This equation may be deduced from the well-known formula 

Qn (<*) = 2^Ti (1 "" ^ 2 ) n ( 5 ~ t)~ n ~ l dt, 
with the aid of the substitution*)* 

When x 9 y and 2 are real the corresponding formula is 

f/ 1 v t 7 ,^ , 

(2 -}- to cosh i/)-* 1 - 1 ai^ = y*"* 1 " 1 ^ (M) ^TTr~ n ~ l P n (u), 
Jo 
where z = /^r. 

EXAMPLE 
Prove that, if a > and | 2z | < 1, 

T 00 (l-2z + it)~ b dt 27rr(a) _ . _ 

I ____2 , ; . '. H* In h . f* 2) 

J _, (1 -f it)++ (1 - it)"-^ 1 2 r (c) r (a + 1 - c) v ' ' ' ; * 

Show also that in the analytical continuation of the integral the line 2z = 1 is generally 
a barrier. 

* Kugdfunktionen, p. 147. 

t Whittaker and Watson, Modern Analysis, p. 319. 



CHAPTER VII 

CYLINDRICAL CO-ORDINATES 

7-11. The diffusion equation in two dimensions. When cylindrical co- 
ordinates p, </>, z are used we have 

x = p cos <, y = p sin </>, 2 = 2, 
and the equation of the conduction of heat becomes 

dv rd 2 v I dv 1 d 2 v 3 

Let us first consider solutions which are independent of z. Writing 
the equation for R is 



- 3 

dp 2 p dp 

ind is satisfied by n 4 r /\x,/3TA/\x 
J B^= A m J m (Xp) -f B m Y m (Ap), 

where J m (Xp) and Y m (Xp) are the standard solutions of Bessel's equation, 
the definitions of which are given in 7-21. In particular, when v is in- 
dependent of (f> the solution is of type 



if A ^ 0, but when A = the solution is of type 

v = A -f- B log p, 
where A and B are arbitrary constants. 

In the case of diffusion from a cylindrical rod r = b to a coaxial cylinder 
which collects the diffusing substance we may use boundary conditions 

such as 

^\ 

v = when p = a, K ~~ = Q when p = b. 

If Q is constant there is a steady state given by 

Q b i a A 

= log - b < p <a. 

K p r 

If there is no rod inside the cylinder but an initial distribution of 
concentration, say v = / (p) when t = 0, we may try to satisfy the conditions 
by a series of type 

00 

v=- S c n e-' t VJ ( /) 5 n ), 

71=1 

where the quantities s l9 s 2 , ... are the different values of s for which 

Jo (a*n) - 0. 



Diffusion from Cylindrical Rod 399 

The solution of this problem is facilitated by means of the formula 

fa 

rJ Q (rs m ) J (rs n ) dr = s m *8 n , 



The solution (I) may be generalised by making A and B functions 
of A and integrating with respect to A between and oo. Many interesting 
solutions of the equation may be expressed by means of definite integrals 
in this way as the following examples will show. 

\ 

EXAMPLES 
1. Prove that if S = p*/4 K t 



/" 

Jo 



-*'' /.(*>)* = e-, 
&<<> 



/ e- x ** cos (A/>) (Ap)-i A* d\ = 

J o 



These are particular cases of Sonine's general formula (Ex. 9, 7-31). 
2. Prove that 



7 (Ap) AdA = 

^w 

where C is Euler's constant. 

/GO 
e~* x2 * J (Ap) ^A = J x /(ir/irf) e~* s / ( 
:> 

-^ 2 7 (Ap) dA = - 

[O. Heaviside, Electromagnetic Theory, vol. in, p. 271.] 

7'12. Motion of an incompressible viscous fluid in an infinite right 
circular cylinder rotating about its axis. Let a> = o> (r, t) denote the angular 
velocity of the fluid about the axis of the cylinder at a distance r from 
this axis, then the equations of motion of a viscous fluid take the form 

dp 

- = /**'. 

9 2 oo 3 3o> 1 3o> 
dr 2 r dr ~~ v fit' 
If the boundary conditions are 

o> = F (r) for t - 0, o> = (J) for r - a, 
the solution given by McLeod* is 



roi = S 2ae/! (JVr) 
=i 

- E ZvNJi (Nr)j (N) e~ N2t (* G (T) e N *'dr, 

n-l Jo 

* A. R. McLeod, Phil. Mag. (6), vol. XLIV, p. 1 (1922). Particular cases of the formula have 
been obtained by other writers to whom reference is made in McLeod's paper. A paper by 
K. Aichi, Tokyo Math. Phys. Soc. (2), vol. iv, p. 2200 (1922) also deals with this problem. 



400 Cylindrical Co-ordinates 

where j (N) J (Na) = 1 and J l (Na) = 0, aN being the nth root of the 
Bessel function. When G (t) = Si = constant and F (r) = 0, we have the 
case in which the fluid is initially at rest and the cylinder suddenly starts 
to rotate with angular velocity Si. 

When G (t) = and F (r) = Si = constant we have the case in which 
the fluid is initially rotating with angular velocity Si and the cylinder is 
suddenly stopped. 

The case in which the cylinder is of finite height 2h may be treated 
with the aid of the equation 

8 2 o> 3 9o> d 2 a) 1 9cu 
dr* + r Jr + ~dz* == vl)i' 

This equation was solved by Meyer* by means of an infinite series of 
type a> = S A n e-*w* </! (ftr), 

where /, ra, k are functions of n connected by the relation 

k* - ra 2 - I 2 . 

A. F. Crossleyt has given a solution of the case in which the boundary 
conditions are 

oj = Si (t) when z = and co = Si (t) when r = a, 

Si (t) being an assigned function of t. This is the case of the semi-infinite 
cylinder. He has also considered the case when a constant couple of 
magnitude G per unit length acts on the cylinder. The corresponding 
problem for an infinite cylinder has been treated by Havelock J. 

Many years ago Meyer applied similar analysis to the problem of the 
damping of the vibrations of an oscillating disc and obtained the following 
relation between the coefficient of damping Jc and the coefficient of viscosity 



where / is the moment of inertia of the disc and n/2 is the frequency of 
oscillation. Kobayashi || has recently made a more exact calculation and 
has obtained a formula 



in which 8 is estimated by means of some approximations to be 2-11. 
Kobayashi's formula, however, does not agree with experiments as well 
as that of Meyer, and the reason for the discrepancy is yet to be found. 
One possible explanation is that the component velocities u and w have 
been ignored. A fuller treatment of the problem has been commenced. 

* O. E. Meyer, Wied. Ann. Bd. xuii, S. 1 (1891). 

f Proc. Camb. Phil. Soc. vol. xxiv, pp. 234, 480 (1928). 

J Phil. Mag. vol. XLJI, p. 620 (1921). 

O. E. Meyer, Pogg. Ann. Bd. cxm, S. 55 (1861); Wied. Ann. Bd. xxxn, S. 642 (1887). 

j| I. Kobayashi, Zeits. f. Phys. Bd. XLU, S. 448 (1927). 



Eolation of a Viscous Fluid 401 

EXAMPLES 
1. If v satisfies the differential equation 

d 2 v 1 dv v d 2 v _ 
ar" 2 + r dr " r 2 + 3? ^ U> 

and the boundary conditions t; = for r = a, v = Fr/a for z = h, v for z = ^, we have 



a 
2. If v = V for r = a, v = for z = A, we have 

^ 4]/ 00 (_)m+l (2m 1) 772 /! [(W A) TTT/k] 

7 ' n=1 m ~ * m [W.Hort.] 

7-13. The vibration of a circular membrane. The equation of vibration 
in polar co-ordinates is 

1 dw 



and is satisfied by 

w = (A n cos &c 4- J5 n sin kct) cos n (<f> + a n ) J n (kr). 

The boundary condition w = for r = a is satisfied if J n (ha) = 0. 
The roots of this equation may be calculated by means of the following 
formula given by McMahon* 

1 4 (4n 2 - 1) (287i 2 -31) 



fc s a p 

where /? == 77 (2n + 46- 1), 

and k s a is the root corresponding to the suffix s in the series k^a, k 2 a, k 3 a, ... 

where the roots are arranged in ascending order of magnitude. 

For the fundamental mode of vibration (n = 0) there is no nodal line. 
For the other modes there are nodal lines which may be concentric circles 
or diameters. The nodal lines for the simple cases are shown in Rayleigh's 
Sound, vol. i, p. 331. 

The solution may be generalised by summation so as to be suitable for 
the representation of a solution which satisfies prescribed initial conditions 

w = w , -g-j = w for t = 0. 

For the determination of the coefficients in the series the following 
formulae are particularly useful. If k and k' are different roots of the 
equation J n (ka) = 0, 

' ^n (kr) J n (k'r) rdr = 0, 
o 



/o 

Annals of Math. vol. ix, p. 23 (1894). See also Watson's Bessd Functions, p. 505. 

26 



402 Cylindrical Co-ordinates 

7*21. The simple solutions of the wave-equation. In cylindrical co- 
ordinates the wave-equation is 

d 2 u 1 du 1 d 2 u d 2 u 1 3 2 u 
Sp 2 ^P dp + p 2 d<f> 2+ dz 2 ^c 2 ~3t*' 
and possesses simple solutions of type 

u = R (p) Z (z) 4> (</>) e lkct = Ve lkct , say 



if H- . , 0, 



A, / and m being arbitrary constants. The last equation is satisfied by 

R = oJ B [p V(* 2 T / 2 )] + 67 m [/> V( 2 T J a )], 

where a and 6 are arbitrary constants. When the lower sign is chosen it 
may be more advantageous to write the solution in the form 

R = al m [p V(l* ~ k*)\ + /3K m \p V(l* - *)], 
where a and /? are arbitrary constants. 

For convenience the definitions and a few properties of the Bessel 
functions are listed below; for a full development of the properties of the 
functions reference may be made to Whittaker and Watson's Modern 
Analysis, to Watson's Bessel Functions and to Gray and Mathews' Bessel 
Functions. The notation used here is the same as that of Watson. 

The function J m (x) is defined by the infinite series 



which converges for all finite values of x. 

When m is an integer we have the relation 

j_ m (x)^(~rj m (x), 

and it is necessary to define a second solution of the differential equation, 
because in this case the two solutions J m (x) and 7_ m (x) are not linearly 
independent. 

The function Y m (x) which gives the second solution is defined by the 
equation 

Y m (x) = lim J -'J&yLll-*to 

v ^ m sin vn 

e-*cos mm 



f ( \ s (1br\ m +* 8 ( 

S ( ,' ' lt ' . , \2 log (\x) 
[9=0 sl(m + s)l \ &V2 

\l 

'\ 

I , 
J 

where y is Euler's constant. 



= -"* 

f 

7/H-8 s } m l /l/j.\2 m p /AM _ o\l 

- - 1 v - ' ^ ' 

-- - ------- 



Properties of Bessel Functions 403 

When x is imaginary it is convenient to use the functions 



&m ( X ) = I 77 " {l-m ( X ) ~ Im ( X )} COS6C W7T = /V_ m (#). 

When m is an integer i m (x) is defined by the equation 

K m (x) - lim K v (x). 

v-*m 

When | arg x \ < \TT and R (m -f |) > the function ^C m (x) may be 
represented by the definite integrals 

rco 

Km ( x ) = e-^cosha (jQg^ ma>f( }a 
Jo 



(00 
(w 2 + a 2 )-* 1 -! cosu.du. 


In the first integral m is unrestricted. 

The functions J w (x) and Y m (x) satisfy the difference equations 



j 

~' 



EXAMPLES 
1. Prove that 






2. Show that when m is a positive integer the relation 

Too 
e im4> I e -kz j (kp) J,m dk = 1 .3 ... (2m - 1) p m 

J o 

may be deduced from Poisson's relatit)ii 

( CO e- kz J (kp)dh = l/r 
J o 

by differentiation. 



26-2 



404 Cylindrical Co-ordinates 

3. Prove that 



kte T - .. 

+ *W_! (p) J n+l (p) + - ( y~ 

r / x r / x 5 ( 5 - 1) r /\T / x 

+ ^m-s+i (p) J n -\ (P) 4 j-g ^ w _s+2 (P) ^n-2 (P) - 

4. Prove that 

i (P) ^n (P) e (wl+n) '*J = 2 ( 2 ^ J^^ ( P ) J n _ s+J) (p) i 



7-22. The elementary solutions involving the function K m (lp) are 
useful for the representation of potentials of distribution of charge located 
near the axis of z. In particular, for a point charge at the origin, we have 
the formula of Basset 



and from this we may deduce the potential of a line charge of density/ (z) 

2 



r 

J-a 



In the neighbourhood of the axis p = 0, this function V becomes 
infinite like 

- 2 1(>KP I dl\ cos / (z - f ) / (f ) df. 

" Jo J oo 

If / (z) is a function which can be expressed by means of a Fourier 
double integral, the foregoing expression becomes 

-2/(z)logp 
at a place where / (z) is continuous and 



at a place where / (z) has a finite discontinuity. This theorem relating to 
the behaviour of the potential of a line charge may be made more precise 
by means of modern results relating to Fourier integrals. The theorem has 
been generalised by Poincare*, Levi-Civitaf and TonoloJ so as to be 
applicable to a charge on a curved line. When the curve is plalie the 
quantity p in the foregoing formula is simply the normal distance of a point 

* Acta Math. vol. xxn, p. 89 (1899). 

f Rend. Lincei (5), t. xvn (1908); Rend. Palermo, t. xxxni, p. 354 (1912). 

t Math. Ann. vol. LXXII, p. 78 (1912); see also A. Viterbi, Rend. Lombardo (2), t. XLII (1908). 



Potential of a Charged Line 405 

from the curve, but when the curve is twisted the expression for p is more 
complicated and the conditions for the validity of the formula harder to 
find. Levi-Civita attributes the asymptotic formula for V to Betti, 
Teorica delleforze Newtoniane (Pisa, Nistri, 1879). 

EXAMPLES 

1. The potential of a row of equal unit point charges with co-ordinates 
z = i/ = 0, z = 27T71 (n = 0, 1, i 2, ...) 

can be expressed in a simple form by adding a compensating uniform line charge on the 
axis of z. The total potential is then 

1 f 
F = M Wdw, 

* J -00 

sinh u 
2u [_cosh u cos z 



where W = ^ | ._.,-""""* - l] 



= - [e~ M cos z -f e~ 2u cos 2z -f ...], 



and t* = (p 2 + w 2 )%. Prove also that 



2 
F = - 2 /C (nr) cos 712, 

W 7^ = l 

and obtain Appell's formula (used in crystal theory by E. Madelung*) 
F = C + - log ( ^ -f ? 2 # (nr ) cos nz, 

7T \ r / 7T n==1 

for the potential of the point charges, C being an infinite constant. This result is allied to 
the general theorem of Lerch [Ann. de Toulouse (1), t. m (1889)], which states that if 8 > 0, 

rr~i r (5 + J) 2 [(z - m) 2 + w 2 ]~ 5 -* = r (5) u~ 2s + 2 2 ^ 

m = -oo 7l= :l 

where ^ n = f e-" 22 - 7 "^ 1 2 s " 1 dz. 

y o 

2. Prove that a particular solution of the equation 



is V = p~ 8 2 K 8 [(p/a) (47T 2 n 2 - a 2 fc 2 )*] f cos ^ (z - 

=i Jo a 

Examine the behaviour of this solution in the neighbourhood of the axis of z. 

7'31. Laplace's expression for a potential function which is symmetrical 
about an axis and finite on the axis. Let us suppose that the potential 
function F is continuous (D, 2) within a sphere S whose centre is at a point 
on the axis of symmetry of the function, then by the theorem of 6*34 
F can be expanded in a power series of ascending powers of the co-ordinates 
x, y, z of a point relative to 0. The fact that F is symmetrical about the| 

\ 

* Phys, Zeit. Bd. xix, S. 524 (1918). Another method of calculating the potentials of periodic 
distributions of charge is given by C. N. Wall, Phil. Mag. (7), vol. m, p. 660 (1927). 



406 Cylindrical Co-ordinates 

axis, which we take as axis of z, means that this power series can be 
expressed in terms of p and z and so is of the form 

V = S a n rP n (p), 

where /A = - . On the axis of z, /A = 1 and P n (p,) = ( l) n . 

Therefore 7 - S a n ( r) n = S a n z n . 

If the value of F is known on the axis and V can be expanded in a 
series of this form, the coefficients a n are known and the function V is 
determined uniquely. Similarly, if we have on the axis 

V = 2 b n z--*, 
the expression for V is given uniquely by 

F=S& n r--ip n Oi). 
Let us suppose that F = / (z) when p = 0. Writing 

F^/(z) + p 2 /2(z)+p 4 /4 (*)+..., 

we find, on substituting in V 2 F 0, that 



where primes denote differentiations with respect to z. The formal ex- 
pression for V is thus 

V =/(*) - /"(*) + -/' v (z)- .......... (A) 

I 
Since ; 

- f' r cos 2n+1 a.da= 0, 

77 Jo 

we find that when /(z -f h) can be expanded in a Taylor series which is 
absolutely and uniformly convergent for a > \ h \ 



1 f"" 

= / ( z 

7T JO 



This is Laplace's expression for the symmetrical potential function 
which reduces to / (z) when p = 0. The formula may be deduced from 
Whittaker's general formula for a solution of V 2 F = 0, namely 

1 f 2 * 
V = f- \ f (z + ix cos a} -f iy sin co, aj) da) (B) 

ATT Jo 

The series (A) may be written in the symbolical form 

V = J ( P D)f (z), 

where D = -^~ . 

vz 

The foregoing method may be applied also to the wave-equation, and 
the formula thus obtained for a wave-function depending only on p, z and , 



Symmetry about an Axis 407 

and reducing to F (z, t) when p = 0, may be deduced at once from the 
generalisation of (B), namely 

1 f 27r 
F = -=- \ f(z + ix cos aj + iy sin to, ct ix sin aj -f iy cos o>, co) dco. 

ZTTJo 

The appropriate formula is 

V -- JP 2 -f ip cos a, sin a r/. 

Z7rJ L C J 

The following special cases of Laplace's formula and the formula just 
given are of special interest : 

1 f 77 " 
r n P n (ju,) == (z -f ip cos a) n da, 

T* Jo 
1 f 77 

r-* 1 -^ (jLt) = - (2.+ ip cos a)-"- 1 rfa, 

77 Jo 

1 f ff 
e- te / (?/>) - - e-"*+v cos a) rfa; 

^ Jo 

^" z '^o [p V(^ 2 + Z 2 )] = o 1 f * e ~ l{z up cosa) - tA ' p 8ln a da ; 
JTT Jo 

the factor e lkct has been omitted from both sides of the last equation. 
The formula 



which holds for both types of Bessel functions, indicates that if U is a 
wave-function independent of </>, then 



is also a wave-function. The effect of this transformation in certain 
particular cas s is indicated in the following table. 

Table I 

U u 



r~ n ~ l P n (p) 

Transforming the expansions of the first expression for U in series of 
Legendre functions and making use of the transformations of the Legendre 
functions indicated by the other two forms of C7, we obtain the expansion 

gin +1 pin 

1.3 ... (2m - 1) - fl) / + 



408 Cylindrical Coordinates 

On the other hand, if we calculate 



by performing an integration by parts after each differentiation we obtain 
the formula 



I yn\ | TTT 

_ 4 j J <* + V cos )-* sin- .<*. 



The corresponding formula for r-*-*P n m (/u,) is obtained from this by 
replacing n by n 1 in the integral. 



EXAMPLES 
1. A solution of the equation 



a? ar 2 "~ ar = > 

which reduces to/() when r 0, is given by the fonrnla 

V = l ["/(* + ir cos 4) (sin ^) 
w ./ o 

where the f unction f(z) is analytic in a rectangle a<$<6, |r|<c;z being equal to 8 + ir. 
2. In the last example if /($) = (a 2 -f fi 2 ) 1 "!", we have 



. 

2 ' 2' 

4. L? (-2)n J^_ F / n+J 1 _^ 2 

^2.4^- l)(n + l)o+ a V 2 J 2' r 2 

-f .... 

3. What problem in potential theory suggests the inversion formula 

1 f* 

$ ( z ) = - I f(z + iacoB w) dw, 

* J o 

1 /*00 Jlf /OO 

f(z} = l I(kti **(-{) 

^ /O -o V fca ) J -oo 

4. If ^ = s 2 4- a 2 , r = 2sa, the equation 

9 2 w> __ B 2 w 2m dw 
dt 2 ~" W + T fo 
is transformed into 

d 2 w 2mdw __ B 2 w 2m dw 

ds 2 r a ~" a? a" ai* 

6. Prove that the differential equation 



is transformed by the substitution 

z = *y, r 

a 2 F a 2 F 

into -5-5- = ^-g 



Integrals involving Bessel Functions 409 

Hence show that this equation possesses a particular solution of type 
V=f*f[xy + cos < V(l - x 2 ) (1 - y 2 )] sin"- 2 <f>d<f>. 

' 



is transformed by the substitution 

x = uv cos </>, y = uv sin </>, z =* J (w 2 v 2 ), 
it is satisfied by V = cos w<.cos nw U (u, v) if 



[P. Humbert, Comptes Rendus, t. CLXX, p. 564 (1920).] 
7. A solution of the equation 

n 2 



~j 

r J 

is given by w = I F [r, 2u \/(*0] e ~ u * du, 

J -00 

where F (r, a;) is an appropriate type of solution of the equation 



d*F n-2dF 
dr 2 r dr' 



8. The solution of + + --- Q, 

ds* dr 2 r dr 



which reduces to 8 V when r = 0, is given by 



9. Prove that if n > 1 and t > 



e~ k l J n (kr) k n + l dk = (2t)- n ~ l r n exp {~ r z /4t}. 

[H. Weber and N. Sonine.] 

7-32. The use of definite integrals involving Bessel functions. The 
potential function represented by the definite integral 



= Pe-toJ (lp)F(l)dl, (A) 

Jo 



in which the function F (I) is supposed to be one which will ensure uniform 
convergence and make the limit of V as p -> equal to the result of 
making p -> under the integral sign, will, when z > 0, take the value 



f(z)= r 
Jo 



on the axis of z, and may often be identified immediately from the form 
of / (z). If, for instance, / (z) = z~ l the corresponding function V is r- 1 and 
the analysis suggests that 



410 Cylindrical Co-ordinates 

This result is easily verified*. Again, if F (1) = J (aZ) where a is an 
arbitrary real constant, we have by the preceding result 



Now this / (2) is the potential on the axis of a unit charge distributed 
uniformly round the circle x 2 -f if = a 2 , z = 0. The function F in this case 
represents the potential of the ring at any point. It should be noticed that 
it is a symmetrical function of a and p. 

In the case of a thin disc of electricity of uniform surface density a on 
the circle x 2 -f y 2 < a 2 , z = 0, the potential may be derived from that of 
a ring by integrating with respect to a between and c. The function 
F (I) is consequently f 



For a circular disc with dipoles normal to its plane and of strength m 
per unit area, the function F is obtained by differentiating with respect 

to z. It is consequently /7 , , , , 7 
1 J F (1) - 2rrmcJ l (cl). 

Similar formulae involving Bessel functions may be used for the 
representation of wave-functions. The natural generalisation of (A) is 



where the lower limit a is at our disposal. Introducing a new variable 
s (I 2 -4- k~}* this becomes, with a suitable choice of a, 



-00 

= e^ 2 -* 2 >i/ n (s p ) f (s) . sds . (,s 2 - 
Jo 



When / (.9) ~ 1 the formula gives Sommerfeld's representationj of the 
function e lkj , which has been used so much in studies relating to the 

propagation of Hertzian waves over the earth's surface. The upper or 
lower sign is chosen according as z ^ 0. 

A wave potential may often be expressed in the form of a definite 
integral involving Bessel functions by making use of Hankel's inversion 

* See Watson's Bessel Functions, p. 384. 

t This result is obtained in a direct manner by A. Gray, Phil. May. (6), vol. xxxvin, p. 201 
(1919). 

J Ann. d. Phys. Bd. xxvm, S. 683 (1909). The formula was given, however, without proof 
in an examination question, Math. Tripos (190.")). See Whittaker and Watson's Modern Analyst^ 
Ewaid, Ann. d. Phys. Bd. LXIV, S. 258 (1921). It was given by II Lamb in a study of earthquake 
waves. Roy. Soc. London. Trans, v. 203 A. pp 1 42 (1904). 



H ankel' s Inversion Formula 411 

formula which holds for an extensive class of functions*. For continuous 
functions satisfying certain other conditions the inversion formula is 



/ (x) = T 

./O 



J m (xt) g (t) tdt, g (t) = J m (xt)f (x) xdx. 

JO 

The inversion formula seems to be applicable in the case of the function 
which occurs under the integral sign in SommerfekTs formula and gives 
the equation f 

I J (Xp) e^pdp/r = (A 2 - &2)-J e - 
J o 



which has been used by H. LambJ in some of his physical investigations. 



EXAMPLES 



/oo 
J m 
> 

F(y) = y~* f 'V*** * m+1 f (*) dx, 
J o 



ro+i roo 

= *T 2 / e~ yx x 
Jo 

prove that under suitable conditions 



so that the relation between the functions / and g is a reciprocal one. 

2. Prove that if the real part of v f 1 is positive the equation 

/(*)= 
is satisfied by f(x) - x v+ c~ x \ [S. Ramanujan.] 

3. When v is subject to the further restriction that its real part is less than 3/2 the 
equation of Ex. 2 possesses a second solution 



[W. N. Bailey, Journ. London Math. Soc. vol. v, p. 92 (1930).] 

* Proofs of the formula are given in Nielsen's Handbuch dtr Cylinderfimktionen, Gray and 
Mathews' Bessel Functions, and Watson's Bessd Functions. New treatments of the relation have 
been given recently by E. C. Titchmarsh, Proc. Camb. Phil. Soc. vol. xxi, p. 463 (1923), and by 
M. Plancherel, Proc. London Math. Soc. (2), vol. xxiv, p. 62 (1926). An extension of the formula 
is given by G. H. Hardy, Proc. London Math. Soc. (2), vol. xxm, p. Ixi (1925), (Records for June 12, 
1924). See also R. G. Cooke, ibid. vol. xxiv, p. 381 (1926). 

f For this equation see N. Soninc, Math. Ann. vol. xvi (1880). 

J Proc. London Math. Soc. (2), vol. vn, p. 140 (1909). 



412 Cylindrical Co-ordinates 

4. A potential which satisfies the conditions 

d 2 F dV 



V - f or z = oo and for J = 0, 
7 = 0/ (p) cos w ^ for = 0, z = 0, 



V 2 F = 0, 
is given by the formula 



V = gcoa m<t> f e~ to </ m (kp) sin (0-*) JfcjF (fc) dk/a, 
J o 



where F (k) = / (a) 7 m (jfca) a da 

7 o 
and a 2 = gk. 

This result is useful for the study of waves caused by a local disturbance in deep sea 
water. [K. Terazawa.] 

5. If the normal pressure on the infinite plane surface of a semi-infinite elastic solid 
is f(p) when p < a and is zero when p > a the normal displacement w is given by the 
formula 

2/iW = - z I e-K* J (kp) F (k) dk-(l + /*/,/) [ e~^ J (kp) F (k) dkjk, 

Jo Jo 

where v = A -f /* and 



o 
If u and v are the lateral displacements 



= - pz f e~ fcz J x (jfcp) JP (A) ^ + (pp/v) ( e~ fe /! (jfcp) ^ (jfc) rfifc/ifc. 

Jo Jo 



2/u (ux + vy) 

[H. Lamb, Proc. London Math. Soc. (1), vol. xxxiv, p. 276 (1902); K. Terazawa, Phil. 
Trans. A, vol. ccxvn, p. 35 (1916).] 

7-33. Another useful formula 
27r/( r >0)= Fudu T rf(p,d>)J (uR)pdpd<f>, R* = r* -f /o 2 - 2r/>cos(0-<i) 

JO J-trJo 

was first given by Neumann. It is proved by Watson* under the following 
conditions : 

(1) It is required that/ (r, 0) should be a bounded function of the real 
variables r and 6 whenever n < 6 < TT and < r. 



(2) The integral f f / (p, <A) o* 

J-7T JO 



is supposed to exist and converge absolutely. 

(3) / (r, 0) considered as a function of r, is required to be of bounded 
variation in the interval (0, oo) for every value of lying between TT, 
this variation being an integrable function of 9. 

* Bessel Functions, p. 470. Less stringent conditions have been discovered recently by Fox, 
Phil. Mag. (7), vol. vi, p. 994 (1928). 



of a Green's Function 413 

(4) The total variation F (r, 6) of the function / (r, 0) in the interval 
(r, s) is required to tend uniformly to zero with respect to as s -> r for all 
values of 6 in the interval ( TT, TT) save, perhaps, some exceptional values 
lying in intervals the sum of whose lengths is arbitrarily small. 

(5) When / (p, <f>) is discontinuous at a point (r, 0) the / (r, 0) outside 
the integral is to be interpreted to mean the mean value 

r' cos Q' = r cos + a cos a, 
r' sin 0' r sin -f a sin a, 

where a is small. This mean value is supposed to be finite as a -> 0. 
We have seen that if u (z) is a solution of the equation 



fco 

the definite integral F = u (z) J Q (lp) dl 

JQ 

is frequently a solution of Laplace's equation. We shall now consider the 
result obtained by taking u to be the Green's function for certain pre- 
scribed boundary conditions at the pianos 2 = c. If the conditions are 
u = when 2 = c, the appropriate function is 

, /v rt sinh/(c-z).sinhJ(c + 2 / ) 



sinh/(c z').siuhl(c + z) , 

- c < z < 2 , 



?y 
and if the boundary conditions are ~ = 0, when 2 = c, the appropriate 

function is 

/x rt cosh I (c z) . cosh I (c + 2') . 
= y <*,*)= 2 ------ V_ --L.- Z ' < 2 < c 



cosh Z (c z') . cosh Z (c + z) , 

= 2 - ' - ^^TTTi - - - " - C < z < z 

smh 2lc 

The resulting integrals have been discussed by Fox* with the aid of the 
identities g ^ z /j = e ^ , ^, + ^ ^ ^ )? 



where sinh 2k .h(z,z')= e~ nc cosh i! (z 2') - cosh I (z + 2'), 

sinh 2k . k (z, z'} = e~ 2l cosh I (z - z') + cosh Z (2 + 2'). 
Now the potentials 

Vl = r*(3,zVo(W<B and V 2= f k(z,z')J (lp)dl 
Jo Jo 



414 



Cylindrical Co-ordinate^ 



are such that v l9 v 2 , ~ and -~- 2 are continuous at the plane z = z', while 



say, hence 



TOO 

U = </ (2, 2') J 
Jo 

I 7 - ry(*,*V 

Jo 



are solutions of Laplace's equation which satisfy the boundary conditions* 

dV 
U -= for z = c, ~ = for 2 = c, 

and are such that J7 r" 1 , F r~ l are regular potential functions in the 
neighbourhood of the point z = 2', x = 0, y = 0. By shifting the axis of 
2 to a new position the singularity of U and F may be made an arbitrary 
point (x', y', z'} between the two planes, and U and V then become Green's 
functions for the space between the planes z c. 

Another expression for U may be obtained by the method of images 
and by the formula of summation given in Example 1, 7-22. The result is 



/ = 



1 



da 
_oo s 



. 775 

Sinh 2c 


. 775 

Sm 2c 


,775 77 (Z Z'} 

cosh - cos rt 

2c 2c 


, 775 77 (Z -f- Z') 

cosh -f- cos 
2c 2c 



(5 2 - a 2 + />*). 

Putting 2' = 2 in both expressions for f/ we obtain the relation 



1 f rfa 


. , 775 

sinh -- 


4Cj.ec 5 


4C , 775 772 

cosh + cos 

2c o 



When 2 this becomes 



rfcr 
3 5 sinh 



775 

2c 



'sinh f^cosIiT ) 

\2c / 



* These results are given by Gray, Mathews and MacRobert in their treatise on Besael 
Functions, and are proved by Fox, loc. cit. 



Source between Parallel Planes 415 

Each integral is, of course, equal to the sum of the series 

1 * __ 2 _____ 2 | 

p (p 2 + 4c 2 )i (p 2 + 16c 2 )* (p 2 -j- 36c 2 )i 

obtained by the method of images. The second expression for U is given by 
T. Boggio, Rend. Lonibardo, (2) vol. xlii., pp. 611-624 (1900). 

EXAMPLES 
1. Prove that when 8 is real 



f> 
t sz.K (sp) J I Fcos s .(, 

J -oo 

/ QO 

.Kg (sp) = I F sin s.d, 

J -co 



sn sz. 



where F - [(z - C) a + wrf*. 

2. Prove that 

/cos (sa cos 0) /!L O (#a sin 0) sin 0d0 2 / cos *{. + I cos J.d J, 
J a W o 

and so obtain a verification of Gauss's theorem relating to the mean value of a potential 
function over a sphere whose centre is at the origin. 

/oo 2c 

6"^ sin cA.J (Ap) A~ J ^A = sin" 1 , 

! ?"l + r 2 

where r x 2 = z 2 -f (p + c) 2 , r a 2 - z 2 + ( P - c) 2 . 

[A. B. Basset.J 

4. Prove that the integral in Example 3 can also be expressed in the form 

, c 4- It sin e 
tan" 1 ., , 

z -f- R cos 6 

where E 2 cos 20 = z 2 4- P 2 - c 2 , -R 2 sin 2e = 2cz. 

5. Prove that when /i > and m and n are positive integers 



[E. W. Hobson, Proc. London Math. Soc. vol. xxv, p. 49 (1894). The first formula was 
given by Callandreau (see Whittaker and Watson's Modern Analysis, p. 364).] 



/: 



6. Prove that , , - a2 f 

-oo 2 -f- 18 

= ' 2 < 0. 

[N. 8onine, 3/a/i. ^nn. vol. xvi, p. 25 (1880).] 

7. A conducting cy Under p = a is surrounded by a uniformly charged ring (x 2 + i/ 2 = a 2 , 
= 0). Prove that the potential outside the cylinder is given by 

V = / e~* z J (sp) J (sb) ds I cos sz . K Q (sp) , f . ds. 
Jo n J AQ (a^) 

8. Prove that [ e~ A2 J m (Ap) e m< ^ 6/A - 1 ( x ~ -* y )"*. 

J T \T -\- Z J 

m>- 1. IH. Hankel.J 



416 Cylindrical Co-ordinates 

7-41. Potential of a thin circular ring. We have already obtained one 
expression for the potential of a thin circular ring, but a simpler expression 
may be obtained by the method of inversion. 

Consider first a point P in the plane of the ring. Let the origin be the 
centre of the ring, a the radius, and let OP = r. Let the mass (or charge) 
associated with a line element add of the ring be aad6, then the potential 
at Pis 

ad<f> _ f 2ff oad<f> 
cos ( 



= t^oadO^ f 2 
Jo R Jo 



where is the angle ROP and < the angle RPQ, while .R denotes the distance 
of the point R from P. 

Now jR -f r cos <f> = RN where N is the foot of the perpendicular from 
on PR. We thus obtain the formula. 

f 2 " 

F = a ad(f> (a 2 - r 2 sin 2 ^)~i - 4a# (r/a), (r < a) 
Jo 

where X is the complete elliptic integral of the first kind to modulus r/a. 

When r > a it is convenient to use another formula which will be 
obtained by inversion. This formula is included, however, in the general 
formula for the potential at an external point and this will now be obtained. 

Let C be an external point, A and B the points on the ring which are 
respectively at the greatest and least distances from C. The plane CAB is 
perpendicular to the plane of the ring and passes through 0. Let the circle 
CAB be drawn and let IJ be the diameter perpendicular to AB. Let CI 
meet AB in P, then CI bisects the angle ACB, and we have 

AC _AP 
BC~PB' 

Hence, if AC =* r 19 BC = r 2 , 



Since ISA = 1C A = ICB 9 the triangles IPS, 1BC are similar and so 

I P. 1C = IB 2 . 

This means that C is the inverse of P with respect to a sphere of radius 
IB. The theory of inversion now indicates that the potential at C is 

V - IB V 
*c- I( j VP- 

Now the triangles ICB and AGP are similar. Therefore 



^ 
IC~AC~BC~ 2a " 



Potential of a Ring 417 



Therefore V c = ^ V P 



where E is the total mass (or charge) of the ring. 

For a point Q in the plane but outside the circle we have 

V Q = 2o- f ackfi (a 2 - r' 2 sin 2 0)-* - 4a (a/r') # (a//), 

J a 

where r' = OQ and r' sin a = a. The expression for the integral is easily 
verified with the aid of the substitution r' sin if; = a sin to. 

Another expression is obtained by integrating the four-dimensional 
potential of a circular ring. This is 

W 



o (x - a cos 0) a -f (y - a sin 0) 2 -f z 2 + w 2 

2rracr 

The corresponding Newtonian potential is thus 



1 TOO 

F = - 

T. J -o 



ro 

Wdw = 2acr. 



Comparing with the previous result we obtain the equation 

A' 



Still another expression for the potential has been obtained in the form 
of a definite integral and so we have the formula 



EXAMPLES 
1. The stream-function for a thin circular ring is 

e~ lz J l (l P ) J (la)dl. 



2. A disc carries a uniform charge distribution of total amount m. Prove that at a poinl 
in the plane of the disc the potential is 



V = [B (alp) - (1 - aVp 2 ) K (a/,)], 

TTt* 

where a is the radius of the disc. Show also that the electric field strength is 

F = 4 [#(a/p) 



where K (k), E (k) are the complete elliptic integrate to rr 'xlulus k. 

B 27 



418 Cylindrical Co-ordinates 

7-42. The mean value of a potential function round a circle. Let us 
first consider the four-dimensional potential 

W = \(x x*) 2 -f (if i/,) 2 -f (z z*) 2 -f (w w* ) 2 1" 1 . 

L\ I/ ' \t/ C7 1 / ' V I/ ' V I/ J 

The mean value round the circle x 2 + y 2 = a, z = 0, w = 0, is 

1 f2- 

IT = - - [(^ a cos 9) 2 + (yi ~ a sin 0) 2 -f- z^ -\- w^]~ l d6 
ZTT J 

= t(^i 2 + y! 2 -f Zi 2 + ^i 2 -I- a 2 ) 2 - 4a 2 (^ 2 -f- 2/ 1 2 )]~ i , 
while the mean value round the circle z 2 f w 2 = a 2 , cc ^ 0, i/ = 0, is 

1 f 2ir 

H^ -= ;r- \x, 2 -f ty, 2 + (2, - ia cos 0) 2 + (w+ - m sin 



= [(^'i 2 + y\ + Zi 2 + ^i 2 - a 2 ) 2 + 4a 2 
These two values are equal. 



Now write V = 



= l [ 

7T J-o 



where J? 2 - (a; - ^) 2 -f (y - y,) 2 f (z 

The mean value of V round the first circle is 

_ 1 



Changing the order of integration and making use of the previous result 
we are led to surmise that 



( 2n 

l Jo ^? 4 2/i 2 



- ______ 

27T 2 _. l o ^? 4 2/i 2 "4- (21 -*i cOs 0) 2 |- (^ - m sin 0} 2 ' 

Again changing the order of integration and performing the integration 
with respect to u\ we obtain 



1 f 2?r 
= * W 

ZTT J 



- 
v = 

J 
The equation thus indicated, viz. 

f * [(^ - a cos 0) 2 -f (?/! ~ a sin 



= f 

Jo 



- a cos 



may in some cases be established directly by writing down Laplace's 
integral (7-31) for the potential of a uniform circular wire and recalling the 
fact already noticed in 7-32 that this potential is symmetric in p and a. 
This equation tells us that if a potential function V (x, y, z) arises from 
a finite (and perhaps an infinite) number of poles, its mean value round 
a circle x l -f y 2 = a, z = 0, which does not pass through any of the poles, is 



= - 1 I' 
TTJQ 



Equation of changing Type 419 

When the circle, round which the mean value of V is desired, lies on 
a given sphere of radius c, it is useful to consider the case in which V can 
be expanded in an absolutely and uniformly convergent series 

V = S (c/r)"^S n (6, ), r > c. 

n-0 

Using our theorem to find the mean value of V round the circle 

x 2 + 7/ 2 - a\ z^b^ V(c 2 a 2 ), 

we notice that S n (6, <f>) is constant on the axis of z, while the integral 
of type l (* ( c V +1 

TT J \ft-f- i cos i/j) 
is equal to P n (6/c) == P n (/x). Hence the mean value is 

F = S PnMSnWoito), 
w-0 

where $ w (0 , </> ) is the value of S n (0, </>) on the axis of the circle, and 
p, = cos a, where a is the angle which a radius of the circle subtends at the 
centre of the sphere. This agrees with the result of 6-35. 
Since at points on the sphere 

v = I: s n (e, <), 

w-0 

we have a means of finding the mean value of a function of the spherical 
polar co-ordinates 0, </> when this function can be expanded in an absolutely 
convergent series of spherical harmonics. 

7-51. An equation which changes from the elliptic to the hyperbolic type. 
We shall find it interesting to discuss a simple boundary problem for an 
equation 



which is elliptic when r < 1 and hyperbolic when r > 1. Writing x = r cos 0, 
y = r sin we shall seek a solution which is such that V = / (0) when r a, 

dV dV 
and shall suppose that F, -x - and ^ are to be finite and continuous for 

d 2 V d 2 F 

r < a, while the second derivatives ^-^, ^x exist. 

or* cu* 

If f (0) = cos 710 and J n (na) 7^ there is a solution 

J n(nr) mn0 ...... A 

J n (na) 

which satisfies the foregoing requirements. Now if z = z (n) is the smallest 
positive root of the equation J n (z) = 0, it is known* that when n is a 
positive integer z (n) > n and that as n -v oo 



* Watson's Bessd Functions, p. 485. 

27-2 



420 Cylindrical Co-ordinates 

Hence if a < 1 we certainly have J n (no) ^ and there is a single 
solution of type (A), but if a > 1 there is no solution of type (A) if a happens 
to have a value for which J n (na) = 0. 

In the more general case, when 

/ (0) = S (A n cos n0 + B n sin n0), 

n-O 

a solution satisfying the condition V = f (9), when r = a, may be given 
uniquely by the formula 



oo 



F = 2 j Y (4 n cos ne + B n sin n0), 

when a < 1 , but when a > 1 there is considerable doubt with regard to 
the convergence of the series and it cannot be asserted that there is a 
solution of the boundary problem in this case until the matter of conr 
vergence has been settled. 

In some cases the convergence may be discussed with the aid of the 
asymptotic expressions for the function J n (na) when n is large. The form 
of these is different according as a 1 is positive or negative. 



CHAPTER VIII 
ELLIPSOIDAL CO-ORDINATES 

8*11. Confocal co-ordinates. An important system of orthogonal co- 
ordinates is associated with a system of confocal quadrics in a space of 
n dimensions. Let x l , # 2 , ... x n be rectangular co-ordinates relative to the 
principal axes of one of the quadrics, then the family of quadrics is 
represented by the equation 



where r is a variable parameter, and a^ 4- r, a 2 2 -1- T, . . . a n 2 4- r are the 
squares of the semi-axes of a typical quadric of the family. It is supposed 
that each quadric possesses a centre ; the case in which the quadrics are 
not central needs special treatment. 

Let us write n 2 p(] 

TI __ i y x s __ * ( r ) 

* T ..ia;-+-;-0(T)' 

where P ( T ) = (T - &) ( T -&)... (T - &), 

- Q(T)=(T + a 1 2 )(r + o^)...(r + a n 2 ). 
We shall suppose that 

a 1 2 >o 2 2 >a 3 2 >...> a n 2 . 

Forming the product Q (r) F r , and putting T = a, 2 , we obtain the 
equations _ ^ Q , ( _ ^ = p ( _ ^ (< = ^ > %) ...... (A) 

which express the rectangular co-ordinates x g in terms of the ' confocal' or 
' elliptic ' co-ordinates 8 . 

The expression Q' ( a 5 2 ), which is the value of the derivative Q' (r) 
for T = a s 2 , is the product of n 1 factors, thus 

Q' (- of) = (a 2 2 - a x 2 ) (a 3 2 - a x 2 ) ... (a n 2 - a^), 

each factor being in this case negative. In Q' ( a 2 2 ) there is one positive 
factor, namely a x 2 a 2 2 , in Q' ( a 3 2 ) two positive factors, and so on. 

Looking at the formula (A) we now see that ( ) n P ( a s 2 ) is positive 
or negative according as s is odd or even. Also ( ) n P ( oo) is positive, 
hence the roots of the equation P (r) = may be arranged in order as 

follows: - h <f 1 <-fl,<f,...<-a tl -<f.. 

The last root n is the parameter of that ellipsoidal quadric of the 
confocal family which passes through the point (x : , # 2 , ... x n ). The equation 
F r = shows that n quadrics of the family pass through this point, and 



422 Ellipsoidal Co-ordinates 

the foregoing inequalities indicate that only one of these quadrics is 
ellipsoidal. 

When a small element of length ds is expressed in terms of , , z , . . . 
it is found that 

12 



j, 



Now 



Q' (- O 



P(-a m ) 1 __;P'(p) 

-! ' (- 2 ) (<' + ^,) 2 " Q (f.) ' 

Therefore ds* = J L f,-/|^ (df ,) . ...... (B) 

j>-l V (?u) 

Laplace's equation in the co-ordinates t) ^ 2 , ... ^ n is thus 

o (o 

....... (0 



This theorem, which was partially known to Green, is generally 
associated with the names of Lame and Jacobi. The case of n variables 
is considered by Bocher* who gives an extension suitable for the family 
of confocal cy elides. 

Let us now denote m by A. It is clear that Laplace's equation possesses 
a solution which is a function of A only if 



where C is a constant. Hence we have the solution 



This is a particular case of a more general solution, namely 



To verify that this is a solution we notice that if K > I 

{P(r)l*_ dr ___ K _ 

i!H= + of r - 

O f~ 9 ' I "\ S\ / 



* 7e&er rfie Reihenentwickelungen der Potentialtheorie, Ch. m, Leipzig (1894). 



Solutions of Laplace's Equation 423 

Now a Q(^) _j_ = _ a(T) = p:_' 

APU,)^-^) 2 STPTr) P* P' 

5 ' (fp) _ 1 _ Q' 

- 



hence we have to show that when K > 1 



P*- 1 Q' 



- 1 Q'~\ 

i~J 



T== ' 



roo ^ /p<-i\ 

that is, that K , ( -^r--, ) r/r = 0, 

JA dr \Q*-*/ 

and this is true. 

9 2 F 3F. 

When K < 1 we cannot differentiate directly to form A, > for ^ is of 

oA oA 

the form 



(r . A)- JB' (r) dr. 
Therefore 



f oo r fi ~] 

= C/c J^ U {(r - A)- E (r)} - (K - 1) (r - A)- 2 (r)J dr 

P(r) }" - <lr K(K - l) -rTflr^ \{ P( ^>\' - 1 __ 1-1 
Q (r)J VQ (r) (A - r)3 J A ^rfr [{ (rj| VQJT) T - A J ' 



_ 4. 

~ 



There is thus an extra term in ,.,, and we have to prove now that 

2 r 



This is true because 

Q (r) P' (A) (r - A) - P (r) Q (A) 

vanishes to the first order as r -> A. It has thus been proved that the 
function 

F=CJV l)V ^, K >0, ...... (D) 

is a solution of Laplace's equation. 

If we write F T = o> we obtain an integral in which the limits for o> are 
and 1 . Since these are constants and occur to an arbitrary power K in 
the integrand we may expect the integrand to be a solution of Laplace's 
equation for all values of the parameter o>. This is indeed true and the 
result may be stated as follows : 

If T is defined by the equation 

n y 2 

2 - =!-, 
_i a, 2 + T 



424 Ellipsoidal Co-ordinates 

wh$re o> is a constant parameter, the function 

w= - 
- 



is a solution of Laplace's equation. 

The potential function (D) may be used to solve seme interesting 
problems. It may be used, in particular, to solve the hydrodynamical 
problem of the steady irrotational motion of an incompressible fluid past 
a stationary ellipsoid. Green's solution of this problem* was amplified by 
Clebschf and extended to a space of n dimensions by C. A. BjerknesJ who 
also considered some additional types of motion of the fluid and called 
attention to the work of Dirichlet and Schering on the problem. The 
analysis is really a development of the formulae of Rodrigues for the 
gravitational potential of a solid homogeneous ellipsoid and of the early 
work of English and French writers on this subject. Historical references 
are given in Routh's Analytical Statics, vol. TI (1902). 

Let us consider a potential V defined by the equations 



= V. = f -/ 
Jo v 



(u) 
If K > we have at the boundary of F = 



inside F < = " 



t > O \ ^ \ > 

CM (M 

while V 2 F - 0. 

If /c > 1 we have also 

VZF roo 3 fF.>-i F.K-I 




Hence if the volume density p be defined by the equation 

V'F, + ^ P h n = 0, 
where the constant h n has the value 



. 

which is readily deducible from the solid angle determined in 6'51, we 

* Edin. Trans. (1833); Papers, p. 315. 

f Crelle, Bd. LII, S. 103 (1856), Bd. Lra, S. 287 (1857). 

J Oott. Nachr. pp. 439, 448, 82d (1873), p. 285 (1874); Fork. Christiania, p. 386 (1875). 

Correspondence ,wr VlScole Poly technique, t. in (1815). See also G. Green, Camb. Trans. 
(1835); Papers, p. 187. He considered problems in a space of s dimensions for various dis- 
tributions of density and different laws of attraction. 



Potential of an Ellipsoid 425 

may regard F as the potential corresponding to a distribution of generalised 
matter (or electricity) of density 



_ 

... a n h n ' 

In particular, if /c = 1 we have the potential of a homogeneous ellipsoid. 
The Newtonian potential of a solid homogeneous ellipsoid is thus 



where a, 6, c are the semi-axes, and 

Q ( U ) = (a 2 + w) (6 2 + w) (c 2 + u). 
The component forces are represented by expressions of type 



A 

and it may be concluded that these expressions represent solutions of 
Laplace's equation. 

The quantity A is defined for external points by the equation 

__x 2 y 2 z 2 _ ' 

a a +"A + 6> A + c a + A"" J 

and the inequality A > ^K' For internal points the lower limit is zero instead 
of A and we may write 

F = Ip {D - Ax 2 - By* - (7z 2 }, 
where A, B, C, D are certain constants defined by the equations 

TOO du TOO du 

A = ^ 6C J (a^I)WM' B= ba6c Jo W^u) 

TOO du roo du 

C = \irabc - : 7~~~; , D = l-nabc - 7^~~- 
^ Jo (c 2 + u) VQ (u)' * Jo VQ (u) 

The component forces at an internal point (x, y, z) are 



8-12. Maclaurin's theorem. The potential at an external point (x 9 y, z) 
may be written in the form 

fo> dv 
V = Kpabc 

j o 



where o x , 6j , c x are the semi-axes of the conf ocal ellipsoid through the point 
(x, y, z). It is thus seen that the potentials at an external point of two 
homogeneous solid confocal ellipsoids are proportional to their masses. 



426 Ellipsoidal Co-ordinates 

8-21. Hypersphere. When the quadric in S n is a hypersphere 

r 2 ~2i2! r 2 _ 02 

~ 1 ' '^2 ' n 

r 2 

we have jP T == 1 , 

a 2 -\- T 

and the potential corresponding to a density 
CK 



_ 
a 2 



-v'-y. 

a + T/ 



where C is a constant and A -- r 2 a 2 . 

In particular, if /<: = 1, we have when n > 2, 

4/7 9/7 9/7 

F - , ox r 2 -*, (r 2 > a 2 ), F - - a 2 - - r 2 a-", (r 2 c, a 2 ) 
?i (n 2) v y Ti-2 n ' v y 

The total mass associated with the hypersphere is in this case -- 



n (n -P2) 
and so the volume of the hypersphere is 



n (n-~2) ' 
Comparing this with the value 



already found, we find that 



The case in which p == / (r) can be solved quite generally with the aid 
of the formulae 



In particular, if / (s) = s n 

/ OX T7 47 ^" a?M4n 

(n - 2) F = -- - --- , r > a 

v ; m 4 n r n ~* ' 



r 

= 4:7rh n 
n [ 



+ ~, -o- r <- 



- 
m + n m+ 2 



Potential of a Homoeoid 427 

CO 

p= S 

m-( 

will give F = 



A density p = 2 C m r 

m-O 



S 

m-O 



(m 



/t\ 

Therefore p = F S (-) 



where 4s 2 = p?lnh n . 

The quantity /* 2 must, moreover, be such that 

\ 



= 



In the notation of Bessel functions 

n r ( n \ ( 2s \* j 

^ r (~2)(r) J |_ : 
where J (- ) = 0. 



8-31. Potential of a homoeoid and of an ellipsoidal conductor. Many of 
the formulae relating to the attraction of ellipsoids and ellipsoidal shells 
may be obtained geometrically. The theorems will be proved for the 
ellipsoid in n dimensions and an extension will be made of the meaning 
of the word homoeoid introduced by Lord Kelvin and P. G. Tait in their 
Natural Philosophy. The analysis is an extension of that given by Poisson*. 

A homoeoid is a shell bounded by two loci which are similar and 
similarly situated with regard to each other. If one locus is an n-dimensional 
ellipsoid and the centre of similitude is the centre of this locus, the second 
locus is an ellipsoid with the same principal axes. 

Let aj , a 2 , . . . a n be the semi-axes of the internal boundary of the shell, 
a l -f da l , a 2 + ^2 > a n + da n the corresponding semi-axes of the external 
boundary. Let OPQ be a line through the centre of the ellipsoids cutting 
them in P and Q respectively, and let OP = r, OQ = r -f dr. 

Let p and p -f dp be the distances of from the tangent hyperplanes 
at P and Q which are, of course, parallel. Then dp is the thickness of the 
shell at P, and we have 

da* da da n dp dr , 

= 2 = ... -* = - = de, say. 

% a 2 a n p r 

* Mdmoires de rinstitut de France (1835). 



428 Ellipsoidal Co-ordinates 

Since the volume of a solid ellipsoid with semi-axes a l9 a 2 , ... a n * s 



__ 
V ( 7 > _ 9\ = 12 

n (n .) 
the volume of the shell is approximately 




a x a 2 ... a n .dt. 



Let us now imagine the shell to be filled with attracting matter of 
uniform density p, then the total mass of the shell is 

n 
Z" 2 j 

j\l -_ pa a* ... a n . de. 

r (l) 

The importance of the ellipsoidal homoeoid in potential theory arises 
from the fact that the attraction of a thin uniform shell of this type is 
zero at any internal point. This is a simple extension of the theorem 
established by Newton for spherical and spheroidal shells. It is important 
in electrostatics because it indicates at once the distribution over the 
surface of an ellipsoidal conductor of electricity which is in equilibrium. 

Through any point / of the region enclosed by the internal boundary 
of the homoeoid let lines be drawn so as to generate a double cone of small 
solid angle dco and to cut out from the ellipsoidal shell small pieces of 
contents dv, dv' respectively. Let a line sSITt completely enclosed by this 
cone meet the boundaries of the shell in the points ST, st respectively, 

T Q 7? T 7? i 7 1? Tffi 7?' T-f 7?' i /-/ 7?' 

JL LJ .it , JL S JLI> ~Y~ CV-/L } JL JL - JTv j J. v - A\> ~\ CLJK . 

Since parallel chords of the two boundaries of the shell are bisected by the 
same diametral hyperplane, we have dR = dR. Also, when daj is very 
small, dv=^ R n ' l dRda), dv' = R' n ~ l dR'dw, hence 

pdv _ pdv' 

and so the attractions at / of the two small pieces balance. When dc is 
very small we may write dv = dpdS, where dS is the area of a surface 
element; the mass of the element dv is thus ppdedS and so the surface 
density is cr = ppdt, thus 



M?) 



2?r 2 aO . . . a n 



Homogeneous Elliptic Cylinder 429 

The potential of the homoeoid at an internal point is constant. When this 
constant value is known the potential at an external point may be found 
by means of Ivory's theorem as in Routh's Analytical Statics, vol. n, 
p. 102. 

8-32. Potential of a homogeneous elliptic cylinder. This potential may 
be found by direct integration*. Let us take the focus 8 of the cross-section 
as origin, then the polar equation of the section is 

6 2 



r == 



a + cos 

where a and 6 are the semi-axes of the ellipse and 2k is the distance between 
the foci. If a is the density of the line charges from which the cylinder is 
supposed to be built up, the potential of all these line charges is 



fT f/(0o) 

--2a dOA logR.r dr , 

J -IT JO 



where R 2 = r 2 + r 2 - 2rr cos (0 - ), 

the infinite constant in the potential of each line charge being omitted. 
Now if r > a -f k we may write 

log -R - log r - S i 
Therefore F = - o6 



- ^0 COB 11(0- 

i n (n + 2)~f J., (a + A; cos ) w 



2* S- f* ^ 

~ J., (a 

- 1:,^^^= <-> r : ') 

Therefore 



- - 

F = - 27raa6 log r - S (- ) ^ cos n0 

L & ! v ; 7i (n 4- 2) \ n J\rJ J 

r > a + k. 

8-33. Elliptic co-ordinates. Potential problems relating to an elliptic 
cylinder may often be solved by using the elliptic co-ordinates of 3-71. 
These are defined by the equations 

x -f iy = a cosh ( -f if]), x = a cosh cos 77, 
t/ = a sinh sin 77. 

The same co-ordinates are also useful for the treatment of the vibrations 
of an elliptic membrane and the scattering of periodic electromagnetic 
waves by an obstacle having the form of either an elliptic or hyperbolic 

* N. R. Sen, Phil Mag. (6), vol. xxxvni, p. 465 (1919); see also W. Burnside, Mess, of Math. 
vol. xvm, p. 84 (1889), 



430 Ellipsoidal Co-ordinates 

cylinder. A screen containing a straight cut of constant width can be 
regarded as a limiting case of an obstacle whose surface is a hyperbolic 
cylinder. IB terms of the co-ordinates , 77 the equation 



becomes ^-^ -f -w 9 -f a z k 2 V (cosh 2 cos 2 17) = 0, 

c$* orj* 

and there are solutions of type F = X (g) Y (77) if 

d ~ + X (a*k* cosh 2 f - 4) - 0, 



~ -f 7 (A - a 2 P cos 2 T?) = 0. 
cw^ 

The equations to be solved are thus of type 

2 + (+ 166 cos 22) y=0. ...... (A) 

This is known as Mathieu's equation or as the equation of the elliptic 
cylinder. The solutions of this equation have been studied by many writers. 
The best presentation of the results obtained is that in Whittaker and 
Watson's Modern Analysis, Ch. xix. 

A discussion of the zeros of solutions of this equation is given in a 
paper by Hille*. He calls any solution of the equation a Mathieu function, 
while Whittaker reserves this name for the solutions! with period ITT. 
Hille remarks that all solutions of the equation are entire functions of z of 
infinite genus. 

The form of the solution when 6 is very large has been discussed by 
Jeffreys^ who also considers the effect of a variation of b on the positions 
of the zeros. 

Jeffreys has also discussed the equation for X, which he calls the 
"modified Mathieu's equation." The equations satisfied by X and Y are 
found to govern the free oscillations of water in an elliptic lake. 

8-34. Mathieu functions. When 6 = and a = n 2 the differential 
equation (A) possesses two independent solutions cos nz, sin nz, which are 
periodic in z with period 77, where n is an integer. 

* E. Hille, Proc. London Math. Soc. (2), vol. xxra, p. 185 (1925). 

f E. L. Ince has shown that for no value of 6, except 6 = 0, does Mathieu's equation possess 
two independent periodic solutions of period 2n. See Proc. Camb. Phil. Soc. vol. xxi, p. 117 (1922); 
Proc. London Math. Soc. (2), vol. xxm, p. 56 (1925). See also E. Hille, loc. cit, ; J. H. McDonald, 
Trans. Amer. Math. Soc. vol. xxix, p. 647 (1927); 2. Markovic, Proc. Camb. Phil Soc. vol. xxm, 
p. 203 (1926-7). A more general type of equation, due to Hill, has been und by Ince to 
possess a similar property, ibid. p. 44. 

J H. Jeffreys, Proc. London Math. Soc. (2), vol. xxiil, pp. 437, 455 (1925). 



Maihieu Functions 431 

When b ^ we can, for any fixed values of a and 6, define an even 
solution c (z) by the initial conditions c (0) = 1, c' (0) = and an odd solu- 
tion s (z) by the initial* conditions s (0) = 0, s' (0) = 1. 

These two solutions of the equation form a fundamental system and 
are connected by the relation 

c (z) s' (z) - s (z) d (z) = 1. 

When 6 is given there are certain values of a for which c (z) is a periodic 
function of period 2?r. These values are roots of a certain determinant/a! 
equation* 

a _l4.86 86 

86 a- 9 86 ... (B) 

86 a- 25 .. 



There are also certain other values of a for which s (z) is a periodic 
function of period 277. The equation determining these values is obtained 
from the last equation by writing a 1 86 in place of a I -f 86. These 
determinantal equations are obtained immediately by substituting Fourier 
series in the differential equation and writing down the conditions for the 
compatibility of the resulting difference equations. 

There is a corresponding determinantal equation for the determination 
of the values of a for which c (z) has a period IT and also one for the deter- 
mination of the values of a for which s (z) has the period TT. 

Whittaker writes ce n (z) for the even periodic Mathieu function whose 
Fourier expansion has a unit coefficient for cos nz, and writes se n (z) for 
the odd periodic Mathieu function whose Fourier expansion has a unit 
coefficient for sin nz. The functions with even suffix have the period TT, 
those with an odd suffix have the period 2-jr. 

The analysis relating to these periodic functions has been much im- 
proved recently by S. Goldstein j who treats the difference equations by 
a method which has proved very successful in the theory of tides on a 
rotating globej. In the case of an even function with period 2?r 



r=0 

The difference equation which leads to (B) is 

{a _ (2r + 1)2} A ^ -f 86 [A^ -f A^} - 0. 

This shows that, as r -> oo, the ratio 

A 

F ^2r+3 
r ~~ A 

-^2r+l 

* See, for instance, E. L. Ince, Proc. Camb. Phil. Soc. vol. xxm, p. 47 (1926-7); Proc. 
Edinburgh Math. Soc. vol. XLI, p. 94 (1923); Ordinary Differential Equations, p. 177 (1927). 

f Proc. Camb. Phil. Soc. vol. xxm, p. 223 (1928). Another method leading to useful results 
has been used by McDonald (I.e.). 

% See Lamb's Hydrodynamics, 5th od. p. 313. 



432 Ellipsoidal Co-ordinates 

tends to either zero or oo. Now in order that the series (C) may converge 
the limit should be zero and not oo. 

To find the condition that this may be the case we write 



then V r ~i = 

and so when V r -> as r -> oo we have 
86 CV 64b*C r C r 
T ~ l ~~ 1 ~o~CV - 1 ~~ciG~^ - 1 ~a CV+2 "" ' 
Now the difference equation 

(a - 1 f- 86) A l -j- 86 A 3 = 

T7 a - 1 -f 86 

gives F ! = ft , - 

Hence we have the equation 

1 a= 86-{- - 2 *- 3 

for the determination of a. 

For the asymptotic expansions of solutions of Mathieu's equation 
reference may be made to papers by W. Marshall* and J. Dougallf. 

EXAMPLES 
1. If 



.1 * 

the equation 
^ 

becomes 



P, M, " V(/x - 1) (v - l)dpL dp 4 p^ [(/ -!)(/*- 1) (v - 1)]* 9 f 

and possesses simple solutions of type 

u R (p) M (p,) N (v) cos m(/) t 
if Jf?, M , N satisfy equations of type 

P 2 (p - 1) E" + (,jp 2 - p) R' h (Jm 2 + hp + Jfcp 2 ) R = 0. 
The substitution R (p) == p* TO /S (sin 2 0) reduces this equation to the equation 

fj? -f (2w -f 1) cot ^ d . - 4 [A 4- k + Jm (m + 1) - k cos 2 0] S 0, 
aa ctt/ 

for the associated Mathieu f unctions J. [P. Humbert.] 

* Proc. Edinburgh Math. Soc. vol. XL, p. 2 (1921). 

f Ibid. vol. XLI, p. 26 (1923). 

J E. L. Ince, Proc. Roy. Soc. Edinburgh, vol. XLII, p. 47 (1922); Proc. Edinburgh Math. Soc. 
vol. XLI, p. 94(1923). 

Proc. Roy. Soc. Edinburgh, vol. XLVI, p. 206 (1926); Proc. Edinburgh Math. Soc. vol. XL, 
p. 27 (1922); Fonctions de Lame et Fonclions de Mathieu, Gauthier-Villars, Paris (1926). 



transforms the equation (D) into 

1 d 2 U (p ~ fi)(/* - v)(v - p) 



[>v(/x 1) (v 
and there are simple solutions of type 



Prolate Spheroid 433 

2. The substitution 



0, 



if p (p - 1) fl" + (p - 4) ' + (A + kp - JAV) # = 0, 

or ^f 4- (a 4- cos 26 + y cos 40) R - 0, 

aa* 

where p = cos 2 0. This is an extension of Mathieu's equation considered by Whittaker* 
and Incef. 

8-41. Prolate spheroid. When the ellipsoid has two equal axes the 
elliptic integrals of 8-11 can be expressed in terms of circular functions f. 
In the case of the prolate spheroid 

r 2 .-I- ?7 2 z 2 

T + 2= *> c2 >^ 2 > 
a 2 c 2 * 

the potential of the homoeoid is 

M t du ___ 

2 JA (a 2 -|- 11) (c 2 -M*)4' 

-v- 2 4_ ? /2 2 

where , ^ + ^-^ = 1, A > 0. 

a 2 4- A c 2 -f- A 

The integral may be evaluated with the aid of the substitutions 

u = c 2 tan 2 j8, cos j8 = 5, 
and we find that 

T7 M. (c 2 -f- A)* + k , , , ^ 

V = 9l lo 2 ^ * - 1 > * = (c ~ a ) ' 
^^ (c 2 -f A)* A; 

Writing x = w cos </>, y = m sin </>, 

2 + ITU = i COS (^ -f ir)) 9 

we have z k cosh 17 cos = &#/^, say, 

w= k sinh 77 sin - fc [(0 2 - 1) (1 - ^ 2 )]i, 

-^ 4. W * = ] 

A: 2 cosh 2 rj ^ Ic* sinh 2 ^ 

* E. T. Whittaker, Proc. Edinburgh Math. Soc. vol. xxxm, p. 76 (1914-15). 

| E. L. Ince, Proc. London Math. Soc. vol. xxm, p. 56 (1925). 

J The results are all well known. Reference may be made to Heine's Kugelfunktionen, 
Bd. ii, 38; to Lamb's Hydrodynamics, Ch. v; and to Byerly's Fourier Series and Spherical 
Harmonics. 

B 28 



434 Ellipsoidal Co-ordinates 

Comparing this with the equation 



c 2 + A ^ a 2 + A 
we see that we must have 

k 2 cosh 2 77 = c 2 4- A, k 2 sinh 2 77 = a 2 -f A, 
J7 M , cosh 77+1 M , , 77 

7 = log r- 1 - = y log COth . 

2k 6 cosh 77-1 k & 2 

The potential may be expressed in another form by finding the distances 
R, R' of a point P from the real foci S, S' of the spheroid. Since 

OS = OS' - k, 
we have 

R 2 = (z- k) 2 + w 2 = k 2 [(cosh 77 cos - I) 2 + sinh 2 77 sin 2 ] 

= k 2 [cosh 77 cos ] 2 , 
J? = k (cosh 77 cos ), R' = k (cosh 77 -f cos ), 

COsh 77 = --zni , 



. 7? +J8' -f 2* 
" 2* g S i- R r -~~2k* 

It is clear from this expression that V is constant on a prolate spheroid 
with S and S' as foci. On the surface of the conductor R + R' = 2c, and V 
has the constant value F , where 

T7 Jf , c -h k 
V ^2k lo %^k' 

The capacity of the conductor is thus 

2k 



(7 



+ k' 



The lines of force are given by = constant, < = constant, and are 
confocal hyperbolas. 

These results remain valid in the limiting case when the spheroid 
reduces to a thin rod SS' 9 and in this case the potential must be capable 
of being expressed as an integral of the inverse distance along the rod. 
We have in fact M ( k ds 

Mj-k [w^lT-"^]*' 

The line density is thus M/2k and is uniform. 

If we take as new co-ordinates the quantities 0, p,, <f>, where 6 = cosh 77, 
fi cos , the square of the linear element ds is given by the equation 
ds* = dx* -f dy* + cfe* - efe 4- <fe 2 -f tn 2 d</> 2 

= i 2 [(cosh 2 77 - cos 2 f ) (d* -f- dr] 2 ) 4- sinh 2 77 sin 2 .d</> 2 ] 

/32 _ ..2 

* - z 2 



[^2 __ ..2 
T^ 



Oblate Spheroid 435 

Along the normal to a surface 6 = constant, we have 



and so the potential gradient is 

3F l/0-l\*8F M 



, 
~ l 6 ~ ' 



, 8^ k \W^* T0 = ' 
At the vertex of the surface = a the gradient is 

_*/._ n-i-_- M 

k* { } ~" i 2 sinh 2 77' 

while at the equator it is 

- 



. ____ _ 

i 2 sinh 77 cosh 77 " 

The ratio of the two gradients is coth 77, which is the ratio of the semi- 
axes of the spheroid. On the spheroid A = 0, cosh 77 = c/k, the surface 
density of electricity is 

1 dV M 






477 dn 4^0 ( C 2_ 
8-42. Oblate spheroid. In the case of an oblate spheroid 



the potential of the spheroidal homoeoid is 



M f 00 du M . _ l 



where fc 2 = a* c 2 and A is defined by 



Making the substitutions 

x = w cos 0, y = w sin <, 

tu + ^2 = k cos ( -f- irj), 

which give z = & sinh 77 sin = &0/i, say, 

tn - k cosh 77 cos f = k [(0 2 + 1) (1 - /x 2 )]*, 
a 2 -f- A = k 2 cosh 2 77 = k 2 (0* + 1), 
C 2 + A = A; 2 sinh 2 rj = fc 2 2 , 

M 
we find that * F - = cot- 1 0. 

A/ 

At a point on the surface of the conductor A = 0, 6 = c/k, and the 
potential has the constant value 



28-2 



436 Ellipsoidal Co-ordinates 

The capacity of the spheroidal conductor is thus 

G = 




2k 
As c -> this approaches the value . This represents the capacity of 

7T 

an infinitely thin circular disc of radius k. If 6, p, $ are taken as new 
co-ordinates the square of the linear element ds is given by the equation 



Along the normal to the surface = constant, we have 

(P2 _ ,/2\i 
V+ f) ' 



Therefore 

SV 



, 



Thus at points of an equipotential surface d = constant, the gradient 
varies like a constant multiple of (O 2 + /i 2 )~*. The ratio of the potential 
gradients at the vertex and equator of this equipotential surface (which is 
likewise an oblate spheroid) is the ratio of the central radii which end at 
these places. 

The surface density of electricity at a point of the spheroidal con- 

ductor iS , 3 rr M 

"=-i-|- --* "<* + */*')-*. 

4?r dn 4-rra ^ 

In the case of the disc this becomes simply 

M M , 

- 



8-43. A conducting ellipsoidal column projecting above a flat conducting 
plane. The electric potential for this case has been studied by Sir Joseph 
Larrnor and J. S. B. Larmor* in connection with the theory of lightning 
conductors, and by Benndprff in relation to the measurement of atmo- 
spheric potential gradients. It is clear that the potential 

r 00 

V = - z + Az \ du [(a 2 + u) (6 2 + u) (c 2 + w) 3 ]"* 
Jx 

is zero over the plane 2=0, and also over the ellipsoid 

2 2 ** 

* Proc. Roy. Soc. A, vol. xc, p. 312 (1914) 

| Wiener Berichte, Bd. cix, S. 923 (1900); Bd. cxv, S. 425 (1906). 



Conducting Column 437 

if A is defined by the equation 



-co 

-^4 du [(a 2 + u) (b 2 + u) (c 2 + 
Jo 



and A by the equation 



/y.2 f/2 ~2 

jr , y , z i A > o 

2 . \ " 1,2 i ^ "T" ^2 . \ ~" A A #** V. 



At a great distance from the ellipsoid V is approximately equal to z 
and the field is uniform. 

When a and 6 are small compared with c so that the column is tall and 
slender, the Larmors remark that A is small and so the lateral effect of 
the column is on the whole small, though the gradient may be very high 
in the immediate neighbourhood of the vertex. When a = b the gradient 
at the vertex is given by the formula 

_ dV __ 2P _ 
dz crtslog-- J- 2a 2 k 

C fC 

where k 2 = c 2 a 2 . With a = 7, c = 25, k = 24, the value of this ratio is 
*about 11-44, so that the gradient is more than eleven times the normal 
gradient. 

In the case of a hemispherical projection of radius a the potential is 

F - 



where r is the distance of the point (x, y, z) from the centre of the sphere. 
At points of the plane z = we have 



dV _ a 5 
dz ~~ rV 



while at points on the axis of z 



The potential gradient at the top of the mound has three times the 
normal value unity. At a point on the plane where r 2a the gradient is 
7/8, while at a point on the axis where z = 2a the gradient is 5/4. The 
effect of the projection on the force thus dies off more rapidly in a vertical 
than in a horizontal direction, but is always more marked at a given vertical 
distance from the sphere than at a given horizontal distance from the 
sphere. 

8-44. Point charge above a hemispherical boss. Let the point charge 
be at P. Let Q be the image of P in the spherical surface of the boss, R the 
image of P in the plane, 8 the image of Q in the plane. 



438 Ellipsoidal Co-ordinates 

Let a be the radius of the sphere, h the distance OP, then the potential 
at T is 

TP" 

for this expression is zero on the plane and also zero on the hemisphere. 

When OP is perpendicular to the plane the force on P due to the 
charges at the electrical images of P is 

1 a 1 a 1 ah ah , 1 



When A = 2a this expression becomes 



IP, ^--l 

a 2 [9 16 25j 



__L 37 _ 
3600a 2 ' 



This is greater than - ( -^ 2 J , consequently the image force is increased 
by the presence of the hemispherical boss, and an electron would tend to 
return to the boss when acted upon by an external electric field 



sufficiently large to just overcome the image force for a perfectly level 
surface. 

8-45. Point charge in front of a plane conductor with a pit or projection 
facing the charge. By inverting a spheroid with respect to a sphere whose 
centre / is at one vertex, we obtain an idea of the charge induced on a 
plane conductor by a point charge when there is a pit or projection facing 
the charge. 

Let jR be the radius of inversion, PP', QQ' pairs of inverse points. Let 
P be on the surface of the spheroid and let e be the charge associated with 
a surface element containing the point P, then 

e IP' 



We shall regard ~ IP' as the charge associated with the corresponding 

point P on the inverse surface 8. Denoting this charge by e', and making 
use of the relation IQ . IQ' = R 2 , we obtain 

JL v e _ y e ' 
IQ' QP~ WP" 

or /| F o= F '> 

where F' is the potential at Q f due to the charges e' on $'. 



Pits and Projections 439 

Placing a charge RV Q at 7 the surface S' will be an equipotential for 
this charge and the charges e' ori S'. These charges e' are in fact the charges 
induced on S' by the charge at /. The force exerted on this charge at / by 
the charges e' on S f is 

F 0v T D V 

= -S c/P cos a - 



where c is the distance of / from the centre of the charges e, which is in the 
present case the centre of the spheroid, and where M is the total charge 
on 8. The force is thus j^ c y 

R* ' 

Except for the pit or projection facing the point 7 the surface S is very 
nearly plane. At infinity it can be regarded as identical with a plane whose 

distance from 7 is h, where D9 

* 



p being the radius of curvature of the spheroid at 7. Since p = a 2 /c, where a 
and c are the semi-axes of the spheroid, we have 

2a% = cR 2 . 

If now the point charge R V were simply in front of a perfectly level 
conducting plane at distance h from it, the force exerted on the charge by 
the induced charges on the plane would be" equal to the force exerted by 
a charge rv at the optical image of 7 in the plane, and so would be 



Comparing this with the force on the charge when there is a pit or 
projection facing it, we find that the ratio of the two forces is v, where for 
the case of a pit 



The value of this ratio v is given for various values of the ratio . The 

c 

TT 

table also includes corresponding values of the ratio -T- , where H is the 

distance of 7 from the bottom of the pit or the top of the projection as 
the case may be : 



* VI 1 V2 2 3 

v 1 3-14 9-67 

f ' i f 1 2 4 



440 Ellipsoidal Co-ordinates 

8-51. Laplace's equation in spheroidal co-ordinates. The quantities 
8, /z, (f) are called spheroidal co-ordinates. In the case of the prolate 
spheroid, we have 

cosh T], fi = cos g , z = k cos cosh 17, tn = k sin sinh 77, 
and Laplace's equation is 

~ * ( - M , V*F s (0* - 1) - + (1 - ) 



while in the case of the oblate spheroid 

= sinh 17, ju, = sin f, 2 = k sin sinh 17, w = k cos cosh 17, 
and Laplace's equation is 

- ft. (* + ,.-) VF S .(^ + 1) + d - /*) 



Some very simple solutions may be found by adopting as trial solutions 
V = f (0 fi) for the prolate spheroid, 
V = f (0 ifji) for the oblate spheroid. 

It is found in each case that f (u) = - satisfies requirements, and so 

u 

we have the solutions 



for the prolate spheroid, and 

V ^ V - 

e* + ^' 

for the oblate spheroid. 



8-52. Lame products for spheroidal co-ordinates. The equation 

V"F + h'V = 
is satisfied by a Lame product of type 

F = M ( M ) (0) O (<f>) 



and 



} - f (- + i) - rf V - ** ( T i)} e - o. j 






Lame Products 441 

the upper or lower sign being taken according as the spheroidal co- 
ordinates JJL, 0, (f> are of the prolate or oblate type. When h = 0, we may 

Wnte 



M = CP n n (n) + JOg, (ft), 
- JS7P n (0) + FQ n m (6) 

for the case of the prolate spheroid and similar expressions for M and O, 
but the following expression 

- EP n m \i0) -I- ^# w w (id) 

for the case of the oblate spheroid, A* B, C, I), E, F being arbitrary 
constants and P n m (//,), Q n m (p,) associated Legendre functions*. It should be 
noticed that we now require a knowledge of the properties of the associated 
Legendre functions P n m (u) and Q n m (u) for arguments u of types u = cosh rj 
and u = i sinh 77. 

When n and m are positive integers appropriate definitions are 



It has been found convenient to write p n (u) for i~ n P n (iu), and q n (u) 
for ^ n+1 Q n (m), then when n is zero or a positive integer we have the 
expansions 

, (2n)\ ( n n(n-\) n 

W " 



, __vi_ / v / \' v ' 7/ n-4 i L 

"*" ~2 .4 (2/i - 1) (2w - 3) + J > 

(271-1- 1)! I"'"' 1 - 2(2^+3) ?I ~ W ~ 3 

jn_+ 1) (n -f 2)_(n f 3) (n f 4) __ 

^ 2 . 4 (2n -f 3) (2n -f 5) 

If = cot j8, it is f oiuid that 

q Q (0) = j8, ft (&) = 1 - )8 cot j8, 
<?2 (^) = i)8 (3 cot 2 j8 -f 1) - | cot /3. 

The coefficient of /? in the expression for q n (0) is equal to p n (0). We 
also have the expressions 

p (0) = (-)n JL ( S i n )n4 
j. /* \ / \ '/vi!> ' 



ooseo 



which are readily verified with the aid of the differential equations satisfied 
by the functions p n (0) and q n (0). 

* The fact that the Lam6 products depend on the associated Legendre functions seems to 
have been first noticed by E. Heine, Crelle, Bd. xxvi, S. 185 (1843). 



442 Ellipsoidal Co-ordinates 

The integral of Laplace's type for the function P n m (6) is 

P n m (0) = smh T? (n -f m)\ f* h ginh cQg a)n _ m gin2 
n v/ 1 .3 ... (2m 1) (n m) ! J 

It shows that P n m (cosh 77) is positive when rj is positive, for when the 
binomial in the integrand is expanded in powers of cos a and integrated 
term by term the odd powers of cos a do not contribute anything to the 
result, while the coefficients of the even powers are all positive. This 
process gives a finite series for P n m (cpsh 17) with the property that each 
term increases with 77. Consequently, when 77 is positive P n m (cosh 77) 
increases with 77. When m = and 77 = the value of the function is 
unity, hence we have the result that 

P n (cosh 77) > 1, 77 > 0. 

8-53. Spheroidal wave-functions. When the spheroid = constant is 
of the prolate type the equations satisfied by M and are both of the type 



where A, a and m are constants and x = (JL when X M and x == 9 when 
X -= C N ). If X = (1 x 2 ) m ! 2 w, we have the equation 



(1 - a; 2 ) - 2 (m + 1) x 4- {A 2 * 2 -f a - m (m + 1)} w = 0. 
rto 2 ' dx l 

This equation has been discussed by several writers*. The present 
investigation follows closely that of A. H. Wilson f. % 

Solutions are required which are finite for 1 c x < 1 so as to represent 
quantities of type M , and solutions are also required which are finite for 
1 ^ x < , where a is some finite constant. These latter solutions are of 
interest for the representation of quantities of type 0, and for this purpose 
a knowledge is also required of the behaviour of solutions of the equation 
for large values of | x \ . 

We commence with a study of a solution 

w =^a s x s y 

represented by a series containing even positive integral powers of the 
variable x. The recurrence relation 

(s + l)(s+ 2) a s + 2 - {* (s - 1) -f 2s (m + 1) - a + m (m + 1)} a s - A 2 a s _ 2 
gives the following equation for the ratio 

N 8 = a 8+2 fa 89 
(9 + m+l)(8 + m)-a A 2 



__ __ 

(s +!)(*+ 2) (8 + 1) (s -f 2)l\r",_ 2 ....... 

* Niven, Phil. Trans. A, vol. CLXXI, p. 231 (1892); R. C. Maclaurin, Trans. Camb. Phil. Soc. 
vol. xvn, p. 41 (1898); M. Abraham, Math. Ann. vol. in, p. 81 (1899); J. W. Nicholson, Proc. 
Roy. Soc. Lond. A, vol. cvn, p. 43 (1925); E. G. C. Poole, Quart. Journ. vol. XLIX, p. 309 (1921). 

f Proc. Roy. Soc. Lond. A, vol. cxrai, p. 617 (1928). 



Spheroidal Wave- functions 443 

When N s -> 1 as s -> oo an approximate value of N s for large values 
of s is obtained by writing N = 1 2r/s in (B) ; it is then found that 
r = 1 m. Therefore for large values of s the coefficients a 8 approximate 
to those in the expansion of (1 x 2 )~ m when m ^ 0, and of log (1 x 2 ) 
when m = 0. In this case X (x) is infinite for x = 1. 

If N s ~ l is unbounded the series represents an integral function and is 
therefore finite in the range 1 <, x < 1 . Also N s ~ A 2 /s 2 and w (x) ~ cosh \x. 

The condition that Km N s = gives a transcendental equation between 

8->co 

a and A 2 . When this condition is combined with the equation 

N _ __ A! _ 

s ~ 2 (s + m + 1) (5 + m) - a - (s + 2) (s + 1) JV.' 
it is found that 

__ ___ A 2 __ ___ __ 3. 4. A 2 _ 5.6^ __ 

~ (wT+~3)~(w + 2) - a - \m 4- 5) (m + 4) - a - ~(m + 7) (wT"6) ^o- - '"' 
the continued fraction being convergent for all values of A and a. Also 
N Q = a 2 / a o = {m (m -f 1) a}/2, and so the equation for a becomes 

/ n_ l-i^L___- 3. 4. A 2 

m (m + 1) a - ^-^-gj ( m + 2) - a - (m + 5) (m + 4) - a 



_ 

(m~-F7)~(w"-f 6) a "" 
A better form is 

1.2.A 2 /{(m-f 2)(m+3)} 
m (m + 1) - a = - - - - 

1 ~ (mT 2) (m T~3) 

3.4.A 2 /{(m+ 2)(m+ 3) (w + 4) (w + 5)} 



(m -f 4) (m + 5) 

This continued fraction gives an infinite number of values of a approxi- 
mate expressions for which may be readily obtained. When m = the 
first value of a is given by the series 

124 16 

a = _ * ^2 f_ A 4 A 6 4- ;\s _|_ 

which converges very rapidly. To find the second value of a it is ad- 
vantageous to write the relation between a and A 2 in the form 
1.2.A 2 /{(m+ 2)(m+ 3)} a 



m (m + 1) - a (m -f 2) (m + 3) 

3.4.A 2 /{(m -f 2) (m + 3) (m + 4) (m + 5)} 



(w -f 4) (m -f 5) 
and it is readily found that 



444 Ellipsoidal Co-ordinates 

If the series for w contains only odd powers of x it is found that the 
recurrence relation is 

(s -f 1) (s -f 2) a, + , 2 = {(s -f m) (s -f m + 1) o} a 8 A 2 a g _ 2 , 
and a is given approximately by the continued fraction 
/~ , iw~ , o^ __2.3.A 2 /{(m-f-3)(m-f4)} 



(m ~f 3) (m + 4) 

4.5.A 2 /{(m + 3) (m + 4) (m + 5) (m + 6)} 
a 

~^ (m H- 5) (m -f 6) 

Priestley* has discussed the solutions of equations (A) by the methods 
of integral equations. If the associated Legendre functions are defined by 
the equations 



Q n (/x) = limit fr coscc WITT ("cos m-n P (/*) - J^ (n + W | Lj P n - ( 

m-> integer |_ I (U m -}- 1) 

we have 

Pn +m (- ft) = cos [(m -f w) TT] P n (p) - (2/7T) sin [(m + n) TT] Q n w (/LI), 
and so we can construct an even function <f> n m (p) and an odd function 
<An m (M) by means of the relations 

</>n m (p>) = (n, m) cos $(m + n) 7T.P n ~ m (/z) -f (2/7r) 5.Q n w (/x), 
V5r n m (n) = (n, m) sin \(m + n) *n.P n - (^ + (2/7r) c.Q n (/x), 
where 

(n, m) - p ^ ^_ -^- jj, 5 - sin J (m - n), c - cos J (m - n). 

Associated with these functions there is an even solution of the 
differential equationf 



which may be derived by solving the integral equation 

<f> n " (n) - U n Ip) - kW sec WTT [*K n m (p, t) U n (t) eft, 

Jo 

and an odd solution of the differential equation which may be obtained 
by solving the integral equation 

\* K n (^ t) V n m (t) dt. 
Jo 



sec 



* H. J. Priestley, Proc. Lond. Math. Soc. vol. xx, p. 37 (1922). 

t This is the differential equation corresponding to the oblate spheroid. 



Integral Equations 445 

In both of these integral equations the kernel K n m (p, t) is 



The functions U n m (//,) and V n m (/*) are infinite f or /x = 1, but 
(1 - ^)4 Z7 n (p.) and (1 - ^ 2 )i F w - (/x) 

are finite. This is not true when m = 0, the corresponding theorem is then 
that U n (/i)/log (1 IJL) and F n (^)/log (1 /n) are finite f or /* = 1. 
Solutions of the differential equation 

< w + x > - - *'*' 2 + 



which approximate to sin (kh9)/6 and cos (kh6)/0 respectively, as -> oo, 
are obtained as solutions of the integral equations 

(1 -f 2 )'* sin khB = Q (6) - f 6? n w ((9, t) (t) dt, 

J CO 

(1 + 2 )-*cos kh9 = Q (9) - f ' O n m (8, t) Q (t) dt 

J 00 

respectively, where 

f m 2 l~l s i n [*A (^ t)] 

0.- (, -<+!)- - -- 



EXAMPLES 
1. Prove that the integral 

* u // 1 M f c sh M (* + ^) 

cfcsmh (tt^l) -- i_. J -f - 



represents a spheroidal wave-function of oblate type. 

2. Prove also that if F (/*, 0) is a solution of V*F + h 2 F = 0, the integral 

$ = P T ^ (<, - ;A) dAcfte<^M 

represent^ a second solution. 

[J. W. Nicholson.] 

8-54. ^4 relation between spheroidal harmonics of different types. The 
four-dimensional potential of the spherical surface 

x 2 + y* -f w 2 - a 2 , 2-0, 
when the surface density depends only on x 2 + y*, is 



= a 2 [*/ (cos 0) sin 0d0 f ** ^ , 
Jo Jo # 



W 
where 

B 2 = (w a cos 0) 2 -f ^ 2 + (x a sin cos <) 2 + (y a sin sin </>) 2 , 



446 Ellipsoidal Co-ordinates 

and/ (/A) is a function giving the law of density. Integrating with respect 
to <f> and writing = cos 6, we obtain 

w = -Y-^-J u (x, y, Z), 

where U = f 

J-i ["(-. 



4- 



5 2 = x 2 + y 2 + z 2 -f w^ 2 -f a 2 , /> 2 = ^ 2 + 2/ 2 . 

Now when ( JT, F, Z) are regarded as rectangular co-ordinates in a three- 
dimensional space /S, the function U is the Newtonian potential of a rod 
of varying density; we therefore introduce spheroidal co-ordinates f, 77 
defined by the equations 

/ U /7 WS * 

COS f COSh 7) = Z = ^r T- ; - ^r , 

7 2 2 



^r T- 

2a (/o 2 



and we find that 

cos = , cosh 77 = 

Corresponding to the standard potential function 

U = Q n (cosh 77) P n (cos f ), 
we then have the four-dimensional potential 



and a corresponding Newtonian potential 

1 f 
V = - Wdw, 

TT } -ao 

which reduces when p = to the value 

00 dw 



To evaluate this integral we use the expansion 

n / - 

Vn w - 



Potential of a Disc 447 

Now if m is zero or a positive integer, 
dw 



00 dw ( 2aw \ n 
oo MJ \z 2 -f- a 2 -f- to 2 / 

a 2 y+*+Jl.3 _ (2m + 25- 1) 



2.4... (2m -f 2* 
= 0, n = 2m -f 1. 

Hence, when n = 2m, 

7r*r(2m+ 1) 1.3... (2m- 1) / a 2 \ m +t 
2 2m4-i p (2m -f~|j ~~~2T4...2m U 2 + a 2 / 



Introducing the spheroidal co-ordinates 

z = ap,v, p = a V{(! - P> 2 ) (^ 2 + !)}> 
we can say at once that 

T7 1.3... (2m- 1) D 7 N , x 
F = 2 " a 2 .4. ..2m P * (/x) ?2 - W 

is a potential function which takes the value F when /z = 1 ; conse- 
quently we have the formula 

dw r s 2 ~ r w 



1.3 ... 



Since this potential function is obtained by projection from the four- 
dimensional potential of a spherical surface it represents the potential of 
a circular disc whose density is a function of the distance from the centre. 
To find this function we notice that a density/ () on the spherical surface 
corresponds to a density (2/)/() on the disc, where p a (1 

If, in particular, we write/ () = P 2m (), we have 

U = 2Q 2m (Z) when X - 7-0. 
Hence the potential 

T7 1.3... (2m- 1) D , 

^=2- 274 ...2m -- P * </*)*">> 

is the potential of a disc of radius a whose surface density is 

(1 - !)-* ^i 



448 Ellipsoidal Co-ordinates 

ap being the distance from the centre. On the disc itself we have 0=0, 
and since 

4W (0) = i 7 ^ 2.~4T... 2m ' 
the value of F is F , where 

2 1 2 .3 2 _... (2m :L l) 2 

This method can be used to find the potential of a disc whose density 
is (2/)/(), where /()i s an even function which can be expanded in a 
series of Legendre functions in the interval 1 < < 1. Sufficient con- 
ditions for the uniform convergence of the expansion in the whole of this 
interval have been obtained in an elementary way by M. H. Stone, Annals 
of Math. vol. xxvii, p. 315 (1926). The requirements are that 



should exist and that the equation;; 

/(z)-=/(- 1) + ' r dut" f" (v)dv 
J-i J-i 

should be valid. The coefficients in the expansion are supposed to be 
derived from/(#) by the usual rule."; 

For further applications of spheroidal co-ordinates reference may be 
made to the books of C. Neumann*, E. Mathieuf, M. BrillouinJ and 
A. B. Basset, and to papers by F. Ehrenhaft||, K. F. Herzfeld 1 ]}, J. W. 
Nicholson**, R. Jones|f and K. SezawaJJ. 

* Theorie der Elektricitats- und Wdrme-Vertkeilung in cinem Hinge, Halle (1864). 

f Cours de Physique Mathe'malique, Gauihier-Villars, Paris (1873). 

J Propagation de V Electricity Hermann, Pans (1904), Ch. vi. 

Hydrodynamics, vol. ir, Cam bridge (1888). 

|| Wiener Ilerichte, p. 273 (1904). 

H Ibid. p. 1587 (1911). 

** Phil. Mag. vol. xi, p. 703 (1906); Phil. Trans. A, vol. ccxxiv, pp. 49, 303 (1924). 
ft Hnd. vol. CCXXVT, p. 231 (192(3). 
Jt Bull. Earthquake Mesearch hist. vol. n, p. 29 (1927). 



CHAPTER IX 

PARABOLOIDAL CO-ORDINATES 
9*11. Transformation of the wave-equation. If we write 

z + ip = (OQ + tft) 2 , 2 - - a, ft 2 = ft 
so that the transformation is 

z=-o-/5, p=2(-j8)i, 
the differential equation 



2m [1 31T d 2 W _ \_ 

_ _ __. _ 



_ 
__ .__^_ ^ 

becomes 



arid is satisfied by* 

ff = A (a) B (0) e' M , 



if a + (m f 1) , - (h - k*a) A = 

2 v ' da ^ ' 



(I) 



where A is arbitrary. 

When A: = it is simpler to use the variables and /3 as independent 
variables, the differential equations for A and B are then 
d*A 2m -h 1 dA 



= 0, 



2m+ldB 



The inference is that there are simple solutions of Laplace's equation 

of the types 7 

<YF 



, .. n . . . , . 

TO (Aj3 ) cos m (</> - c/> ), 

m (Aj8 ) cos m (j> - ^ ), 
m (Aft) cos 7/1 (</> - ), 
n (Aft) cos m (^ - c/> ), 
where A and m are arbitrary constants. The expressions for p and 2 in terms 

of a and are , 

z-tto 2 -^ 2 , /t>-=2a ft, 

r = a 2 -f ^ 2 . 

* H. J. 8harpe, Quarterly Journal, vol. xv, p. 1 (1878); Proc. (Jamb. Phil. Soc. vol. z, 
p. 101 (1899); vol. xm, p. 133 (1905); vol. xv, p. 190 (1909); H. Lamb, Proc. London Math. Soc. (2), 
vol. iv, p. 190 (1907). 

B 29 



450 Paraboloidal Co-ordinates 

Many well-known potentials may be expressed in terms of these simple 
solutions in an interesting way. We shall give here an expression for the 
inverse distance 

ft~ l = [(* + y 2 ) 2 + pT* 

= K 4 + A, 4 + yo 4 - 2ft V + 2y 2 o 2 + 2o 2 ft 2 r i . 

Let us assume that 



(Ao,,) J (Aft) J (Ay )/ (A) d\, 
o 

then when ft = we should have 

(o 2 + yo 2 )- 1 = jj K (Ao) J (Vo)/ (A) d\. 
This indicates that perhaps 

Q (Aft) /(A) 



o 
and again, when ft = 0, we should have 



= f C 
Jo 



o 
This equation is satisfied by/ (A) = (7A, where C is a constant such that 

C {" K Q (T)rdT= I. 
Jo 

Too 

Now ^L O (T)= e- TC08hu dtt, 

Jo 

Too fcoroo 

and so K^(r)rdr = \ rdr \ e-'coshu ^ w 

Jo Jo Jo 

TOO 

= sech 2 ^.du = 1. 
Jo 

Hence (7=1, and so the analysis suggests the equation 

R-i = f " K Q (Aa ) J (Aft) J (Ay ) A dX. 

Jo 

This equation may be checked in many ways. In particular, if we make 
use of the equation 

77/0 (Aft) J (Ay ) = f ' Jo (ft 2 + yo 2 ~ 2ft y cos o>) do> 
Jo 

the relation may be deduced from the simpler relation 

(o 2 + So 2 )- 1 = f K. (A ) J (AS ) AdA, 
Jo 

which in turn may be checked by substituting the foregoing expression 
for K Q (r). The equation for R~ l is a particular case of one given by 
H. M. Macdonald*. A proof of the formula is given in Watson's Bessel 

* Proc. London Math. Soc. (2), vol. vn, p. 142 (1909). 



Sonine's Polynomials 451 

Functions, p. 412. Some analogous integrals have been evaluated by 
Watson* at Whittaker's suggestion. 

The corresponding expression for -R^ 1 , where 

^i 2 = (* - yo 2 ) + P 2 , 

is jRr 1 = [ </o (Aa ) K Q (A&) J (Ay ) AdA. 

Jo 



9*21. Sonine's polynomials. Putting 2ikn = ik (m + 1) h in the 
equations (I) we find that the differential equations are satisfied by 

A = e~**F m n (2ika), B = e~^F m n (2ikp), 
where F m n (s) is a solution of the differential equation 



This is a slight modification of Weiler's canonical f ormf for an equation 
of Laplace's type. It is satisfied by a confluent hypergeometric series of 

type 



_ n 



1 (m + 1) 1 . 2 (m -f 1) (m + 2) 

which is usually denoted by the symbol F ( n\ m + 1; 5), but in the 
British Association report for 1926 the symbol F (a;y; s) is replaced by 
M (a;y; 5). 

When n is a positive integer a solution of the equation can be expressed 
in terms of Sonine's polynomial;]: , T m n (s), which may be defined by means 
of the expansion 

(1 -f $)-'*- 1 e 1+l =S r (m-f- n+ l)t n T m n (s), ...... (A) 

no 

where | t \ < 1 . Calculating the coefficients in the expansion of the function 
on the left-hand side of the equation we find that 

oTt oft 1 



______ ^__ __ __ __ 

(m + n- l)n- 2! 2! 



if we adopt the modern notation for the generalised Laguerre polynomial. 

* Journ. London Math. Soc. vol. m, p. 22 (1928). 

t Crelle's Journal, vol. LI, p. 105 (1856). 

t Math. Ann. vol. xvi, p. 1 (1880). The T notation was probably adopted in honour 
of Tchebycheff who considered particular cases of the polynomial in 1859; Oeuvres, t. I, 
pp. 500-508 (1899). 

29-2 



452 Paraboloidal Co-ordinates 

When m -f n is zero or a positive integer there is another formula* 

/ _ \m+ngs d m+n 

n ' ~ S " 



which indicates that in this case the polynomial may also be defined by 
means of the expansion 

1LIL^!! C A. = s s X+"T m n (s). ...... (F) 



A comparison of the formulae (D) and (E) indicates also that in this 
CaSC * m T m (s) = T_ m (s). ...... (G) 

With the aid of the last relation many formulae may be duplicated. 
Sonine's polynomial satisfies a number of difference equations, most of 
which are given in the memoir of Gegenbauer|. 

T m -S (*) = (* + m) TV (s) -(m + n+l)s T m+l (s), 
n(m \~n) T m " (s) - {s - (m + 2n - 1)} 7V" 1 (s) 4- TV" 2 (s) = 0, 

7V- 1 (s) ~(m + n) 7V>-i (s) + T ^n-* (s)f 
(n - 1) 7V" 1 (s) =- {s - (m + rc - 1)} T m+1 ~* (s) - T m+l "-* (s), 
(n - 1) 7V- 1 (*) - {s ~ (m + 1)} T m+ ?~* (s) - s T m+ f~* (s), 
s ~ [T m (s)] = T ro n-i (8) + nT m (8), 

(I* 

\T n (^ T n ~v ($} 

m ~~ m ~ 



(m -f- n + 1) ^[TV 1 -* 1 (*)] - -V (^ - j s [T m (s)]. 
From these equations we may deduce that 

' 7 / 



When w (w -f i) is positive or zero this equation shows that 

s-^T^+i (8)/T m (8) 

increases with s except possibly at a place where both T m n (s) and T m n+l (s) 
are zero. The roots and poles of this function consequently occur alternately 
as ,s incre^ises from oo to oo, and so the roots of the equation T m n (s) = 
separate those of the equation T m " } x (.9) = 0. 

Gegenbauer showed that if m ^ 1 the roots of the equation T m n (s) = 0, 
c^ nsidered as an equation of the ?ith degree in .9, are all real, positive and 

* Deruyts, Liege mi moires (2), t. xiv, p. 9 (1888). 
t Wiener Benchte, Bd. xcv, S. 274 (1887). 



Orthogonal Relations 453 

unequal. This is a generalisation of a result obtained by Laguerre* for the 
case m = 0. It is an immediate consequence of an orthogonal relation 
which will be obtained presently. A geometrical proof has been given by 
Bocherf for the case m . The distribution of the roots is of some 
physical interest because Lagrange J showed that the equation T n (s) = 
gives the possible periods of oscillation of a compound pendulum con- 
sisting of equal weights equally spaced on a light string. The transition 
from the compound ' pendulum to a continuous heavy flexible chain has 
been discussed by Suzuki . Properties of the roots are given by E. R. 
Neumann |! . 

9-22. Orthogonal properties of the polynomials. Let us consider the 
integral 

I n v = [^ e~* s sT m " (as) T m * (bs) ds, a > 0, b > 0. 
h 

If | t | < 1 and | r | < 1 we may write* 
2 2 T (m + n -f 1) T (m + v + 1) t n r" I n%v 

n=Q ^=0 

= I e-**s m ds.[(l + *)(! + T)]-- 1 exp 

J 



-I t 1 + 

x-a)t+ (x-b)r+ (x-a- b) 



x [a; + (x - a) J]--"- 1 . 

The coefficient of ^ n r" in this expansion is easily obtained, and we 
find that^j 

j _ , _ (^y^ v r _( m +_ n + v JL_ 1 L _ ( x ~ a ) n ( x ~^ v 

n >"~ T (n+ 1) rliT+""l)T7m~+"w + 1) f (m + v + f) x w+ +''+ r ~ " 

n / ( a 6) 

x ^ n, v\ m 7i v\ . .-. /r 
V (x - a) (x - b) 

with the usual notation for the hypergeometric function. 
When x == a = b = 1 the double series has the sum 
T (m + 1) (1 - tr)-- 1 (m > - 1) 
and so we find that in this case 






(+l)+ m 4- 1) 

. /Soc. J^a^. de France, t. vn, p. 72 (1879); Oeuvres de Laguerre, t. i, p. 428. 

t Proc. Amer. Acad. of Arts and Sciences, vol. XL, p. 469 (1904). 

t Miscellanea Taurinensia, t. m (1762-1765); Oeuvres, t. i, p. 534. 

Proc. Phys. Math. Soc. Japan, vol. n, p. 185 (1920). 

|| Jahresbericht deutsch. Math. Verein. Bd. xxx, S. 15 (1921). See also a paper by A. Milne, 
Proc. Edin. Math. Soc. vol. xxxm, p. 48 (1915). 

If This is a simple generalisation of a formula given by P. S. Epstein, Proc. Nat. Acad. of Set. 
vol. xn, p. 629 (1926). 



454 Paraboloidal Co-ordinates 

This orthogonal property of the polynomials was discovered by Abel 
and Murphy for the case m = 0, and by Sonine for the general case. In the 
cases m = J, Sonine's polynomial can be expressed in terms of Hermite's 
polynomial U n (x), which may be defined by means of the equations 



We have in fact m . 9 . , /ft . , rr . . 

\= U 2n (x), 



.(*) 
This polynomial possesses the orthogonal property 

roo 

e U m (x) U n (x) dx = (m * n) 

J -00 

= 2 n (n!) (m = n), 
so that the functions U n * (x), defined by the equation 

f/n* Or) = (2n!) -'<-<">'> f/ n (x), 

form a normalised orthogonal set for the interval (00,00). It seems that 
these functions first occurred in Laplace's Theorie Analytique des Pro- 
babilites. 

In papers on the new mechanics Hermite's polynomial is usually 
denoted by H n (x) instead of U n (x). A useful bibliography on Hermitian 
polynomials and Hermitian series is given in a valuable paper by E. Hille, 
Annals of Math. vol. xxvn, p. 427 (1926). 

EXAMPLES 

1. Prove that n ! \ e- X8 8 m T m n (a) ds = (1 - x) n ar rw ^ n ~ 1 . [N. Sonine.] 

J o 

2. Prove that 

r (m + %).T m n (s) I T, n (s cos 2 ^).sin 2m *t>.d<f> (m > - J). 

Jo f 

3. Obtain also the more general formula- 

r (p) T v+p n (x) - / T v n (xy) y v (\- y)r~ l dy 
Jo 

(p > 0, v > - 1). [N. S. Koshliakov*.] 

4. Prove that 

r (n + v + 1) *+M+I T^^ 1 ^" (a) = r (n + 1) r (y + 1) f% - t) m fr T m n (s - T* (t) dt. 

J o * 

5. Prove that T m (x) - j^ ^ !T m *-* ^ . [B. M. Wilsonf.] 

6. Prove that e -x (i VM)-<" J m (2i VM) - 1 x n T m n (z). [N. Sonine.] 

* M. o/ AfflM. vol. LV, p. 152 (1925). 
t Ibid., vol. uii, p. 159 (1924). 



Expansion of a Product 455 

7. Prove that, if m and n + 1 are positive integers and 8 > 0, 

I T m n W | < n ! m ! eK [G. Szegd.] 

8. Prove that the equations * 

cPx 
jp = n (*n-i - *n) - (n + 1) (a? w - x n+l ) 

have a particular set of solutions of type 



where L n (u) = 1 - + - ... + (-)" . [J. L. Lagrange.] 



9. Prove that e"rt - (I - ) 2 s n n (u), 

n=0 

and that consequently L n (u) is the so-called polynomial of Laguerre which possesses the 
property 

cr"L m (u)L n (u)du 

=sO m ^ n 

= I m = n. [E. T. Whittaker.] 

9-31. An expression for the product of two Sonine polynomials. The 
analysis of 921 indicates the existence of wave-functions of type 



W=~00 71=0 

where the coefficients A m n are suitable constants. 

The convergence of a series of this type may be partially discussed 
with the aid of the equation* 

n (m -f n + 1) x 2 



(^ - 1) (m -f- n + 1) (m -f n + 2) 



"1 
"J ' 



1 . 2 (m -f I) 2 (m -f 2) 2 (m + 3) (m + 4) 

in which the coefficients on the right are all positive. This equation, which 
will be established presently, shows that the modulus of T m n (ix) increases 
with x 2 . Hence if the series (A) converges absolutely for any given value 
of | a | , it converges absolutely for all smaller values of | a \ . 

This equation may be established by first expressing a typical term in 
the expansion for i2 as* a definite integral of type 

r2ir 

f[z ix cos o> iy sin to, ct x sin o> 4- y cos oo, co] da). 
Jo 

Taking 

f = e tfc[-c<-t(-iv)e iw ]-ima, F [z ix cos o> iy sin co], 

* An analogous equation for the confluent hypergeometric function F(a; y\ s) was ob- 
tained by S. Raman uj an and extended to the generalised hypergeometric function by 
C. T. Preece, Proc. London Math. Soc. (2), vol. xxn, p. 370 (1924). The present equation was 
given in the author's Ekctrical and Optical Wave Motion, p. 101 (1915). 



456 Paraboloidal Co-ordinates 

we may write the integral in the form 

f2ir 

elk(z -ct)-im4> e&SV-imy F \Z - lp COS y] dy 

Jo 

by making the substitution y = o> <f>. Our aim now is to choose the 
function F, so that 

(kp) m T m n (2ika) T m n (2ikp) = f %*>''*-<*> f (z - ip cosy) dy. 

Jo 

We shall verify that a suitable function F is given by 



F (s) = -~ ------ 3P (- 2O). ...... (B) 

v ' 2?r.r (m + n + 1) v ; v ' 

To do this we multiply both sides of our equation by e~**, where is 
an arbitrary positive quantity greater than p, and we then integrate 
between and co. Since each side of our equation is a polynomial in k 
the equation will be verified if it can be shown that the resulting equation 
is true. 

Making use of the formula of 9-22 the resulting equation is found to be 

|*2ir 

e -tmy (^ + pe -iv)n ( 

Jo 

_ r(ffM-2n-|-l)27r 
" T"(n V 1) r (m -f n -f 

where rj = ^ -f 2iz. Now, if and 77 are sufficiently large we find by 
expansion in powers of p that the definite integral has the value 



Putting u = p 2 /^, the equation to be established is 
|T(m-frc4- l)] 2 ^(m-f n-h 1, - n;m+ 1;^) 
= T(m + 2w+ 1) T(m+ 1)(1 - u} n F[-n,-n\-m- 2n;(l- w)- 1 ], 

but this is true on account of a well-known property of the hypergeometric 
series. 

Putting jS = , p = 2a, our formula becomes 

27r T (m + 7i-fl) (-) n (2ak) m T m n (2iak) T m n (- 2iak) 

(2n 

= I exp [2ake*y - imy] .-T n (- ^aA: cos y) dy, 

and from this equation the expression (B) is readily derived. If we write 

e*> == r, - 2ak = s\ 
the foregoing equation gives the expansion 

e~* T T n [s (r + r- 1 )] - 2 (-)+ r (m + n + 1) (rs)" T m * (is) T m (^ w). 



Confluent Hypergeometric Function 457 

This is a particular case of a more general expansion 
exp [(,)* &"} TV [f + 7, - 2 (,)* cos o>] 



= S (-) T (m + n + 1) (,)/ " 2V (0 T m (T?). 

m= n 

The differential equation (for F) has been studied for general values of 
m and n by Pochhammer, Jacobstahl, Whittaker, Barnes and many other 
writers. Writing it in the canonical form 



where a and y are arbitrary constants which may be complex, the complete 
solutionis 



where F (; y; *) - 1 + ^ x + - (} * + - 

is the confluent hypergeometric series which is so named because we may 
write* . N 

F(a;y;x)= lim ^(;;y; A 

/g^oo \ P/ 

When y is a positive integer the coefficient of B is either infinite or 
identical with the coefficient of A. In this case the complete solution 

of (C) isf / 7 _ 2 

y=[A + Clogx]F(a;y;x)+c'2Ji-)+y B ( w +- " 

i 



, o _ _ i _ ^ __ 

r \ y / y(r+ ij2iU" 1 "a+ i y y+i 2 



_ __ __ __ __ 1 __ 

y (y -f l)(y+2) 3! \a -f l^a + 2 y y ~+ 1 y -f- 2 2 3 

+ ... to infinity. 
When m and n are positive integers the equation 



possesses only one solution T m n (5) which can be represented by a con- 
vergent power series of integral powers of s. A second solution may be 
derived by a well-known method by writing F = uT m n (s). The equation 
for u is then 



and so F = CT m n (s) + DT m n (s) f * e [T m (<r)]- 2 <,-(+ da. 

* E. Kummer, Crdlea Journal, vol. xv, p. 138 (1836) 

t H. A. Webb and J. R. Airey, Phil. Mag. xxxvi, p. 129 (1918). W. J. Archibald, Phil. Mag. 
a. 7, vol. 26, pp. 415-419 (1938), (addition of the second term for y > 1). 



458 Paraboloidal Co-ordinates 

This solution gives a logarithmic term when the integrand is expanded 
in powers of a. 

When s is imaginary, or a complex quantity, as it is in our case^ use 
may be made of the solution represented by the definite integral 

Z7 -" {s) = rWT 

A number of analogous definite integrals are given in papers by 
Epstein* and Whittakerf. 

Whittaker reduces the differential equation to a standard form 



dz* ' ( 4 ' z ' 2* 
and introduces as the principal solution the function 



1 f (0+) , / <\*-4+ m 

W k> m (z} = - r (k + | - n) e-l* z" J^ (- t)- k -* +m (l + I) e-' dt, 



where arg z has its principal value and the contour is so chosen that the 
point^ = z is outside it. "The integrand is rendered one-valued by 
taking | arg ( t) \ < TT, and taking that value of arg (1 -f t/z) which tends 
to zero as t -> by a path lying inside the contour." When 

R(K- |- m)< 0, 



"This formula suffices to define W km (z) in the critical cases when 
ra -f k \ is a positive integer, and so W km (z) is defined for all values of 
z except negative real values." 

The relation between this function and Sonine's polynomial is indicated 
by the relation 

- 



_ /TT n 

~ 



n ! r (m + n + 1) 

For the properties of the function PT fc>m (z) and its asymptotic expansion 
reference must be made to Modern Analysis and to some later papers by 
mathematicians of the Edinburgh school J. The asymptotic expansion of 
the function T m n (x) for large values of n is discussed by J. V. Uspensky 
and used to obtain sufficient conditions for the validity of the expansion 
of /(z) in a series of these functions when the coefficients are given by 
means of the orthogonal relation of 9-22. The summability of the series 

* Diss. Munich (1914). 

f Bull. Amer. Math. Soc. vol. x, p. 125 (1904). 

J D. Gibb, Proc. Edin. Math. Soc. vol. xxxiv, p. 93 (1916); N. M' Arthur, ibid. vol. xxxvin, 
p. 27 (1920); G. E. Chappell, ibid. vol. XLIH, p. 117 (1924). 

Annals of Math. vol. xxvm, p. 593 (1927). See also 0. Perron, Crelk, Bd. CLI, S. 63 
(1921); M. H."Stone, Annals oj Math. vol. xxix, p. 1 (1927). 



Examples 459 

has been discussed by E. Hille* and G. Szegof. The Parseval theorem for 
the series has been investigated by S. WigertJ and M. Biesz. 

EXAMPLES 

1. Prove that 

e^^^J m (kpaina))^ 2 A m n p m e ik * T m n (2ika) T m n (2ikp), 

n=0 

o> / oA^H-Si 
where A m n - (-) n r (n + 1) r (m + n + 1) k m sec 2 ^ f tan^ J 

2. If (x) is a continuous function for all real values of x and if, when | x \ is very 
large, <f> (x) (e~ ka ^) 9 where k is a positive constant, the equations 



00 

w-0, 1,2,..., 

in which H n (x) denotes Hermite's polynomial, imply that </> (x) for all real values of x. 

[M. H. Stone.] 

3. Iff(x) is integrable for all real values of x and such that 



exists, the quantities 



are such that the infinite series 

2 2 n n!c n 2 

n=0 

converges. [M. H. Stone.] 

4. If, in addition, the limits/ (x 0) exist absolutely and / (x) is absolutely integrable 
over any finite interval, the series 

2 c n //(*) ...... (D) 

n=0 

converges and its sum is [/ (x -f 0) -f / (2*0 ~~ 0)]. [J. V. Uspensky.] 

5. If / (x) admits the representation 

f'(z)dz (-00 <*<oo) 

o 

and f e~* [2xf (x)-f' (a;)] 2 rfa: 

J -co 

exists, and if, moreover, for large values of | x \ 



the series (D) converges uniformly tof(x) in any finite interval. {M. H. Stone.] 

* Proc. Nat. Acad. Sci. vol. xn, pp. 261, 265, 348 (1926). 
t Math. Zeits. Bd. xxv, S. 87 (1926). 
t Arlcivfor Mat., Astron. och Fysik, Bd. xv (1921). 
Szeged Acta, 1. 1, p. 209 (1923). 



460 Paraboloidal Co-ordinates 

6. Prove that, if m > i, we have for large values of n 

n n (x) = n (x) fcos n 4- (ns)-*tt (a) sin 6 n -f i n (* 

n n ' (a?)- - o> (x) j~(n/*)* sin n - i v (x) cos n + (nx)~* ^n 
where n n (a) = (-) n [r (m + n + 1) r (n -f 1)]* ZV (*), 

)-*, ^ n = 2v/^"- 4 ^ TT, 

, . x 2 1 -f m 1 m 2 

w (^)=12--2-^ 16" 4 ' 

, . a: 2 mx 1 (m -f I) 2 

"^-ii-T + ie- 1 -^' 

and where (a:) and ^ (x) remain bounded when x varies in the finite interval 

< a ^ x ^ b. 

In this form the asymptotic expression is due to Uspensky. An earlier form, given by 
Fej6r, has been elaborated by Perron and Szego in papers to which we have already 
referred. The corresponding asymptotic expression for Hermite's polynomial was obtained 
by Adamoff, Ann. lust. Polytechnique de St P&ersbourg, t. v, p. 127 (1906). The result 
was extended to complex values of the variable x by G. N. Watson, Proc. London Math. 
Soc. (2), vol. vra, p. 393 (1910). 

7. Prove that s T m+1 () = (n + 1) T m n ^ (s) -f- T m " (s). 

8. Prove that, if a > 0, 



J-C; <; 2 )^- r(a 2 : i 1 - ) c) ^ ) / 



l + ,)- (i-is 



oo (Ji/r} / 1 \ 

9. Prove that JF (a; y; - h) e** = 2 ^ ^ 1 a, - m; y; ) . 

7n=,0 ml \ X) 

[P. Humbert, Journ. de Vficole Polytechnique (2), Cah. 24, p. 59 (1924).] 



CHAPTER X 
TOROIDAL CO-ORDINATES 

10*1. Laplace's equation in toroidal co-ordinates. If we put 

x = p cos <, y = /> sin </>, 2 -f i/o = a cot \ (if/ -f ia), 
a sinh a a sin 



- - __ 2J _ .. 

cosh a cos i/r ' cosh a cos $ ' 

the angle is the angle subtended at a point P (x, y, z) by the segment 
A B which is the diameter of the circle z = 0, x 2 -h y 2 = a 2 in the plane 
through P and the axis of z. The quantity a is equal to log (PB/PA). The 
surfaces \f> = constant are the spherical caps having the above-mentioned 
circle in common; the surfaces a= constant are anchor rings. In these 
co-ordinates 

dx 2 -f <fy 2 -f <fe 2 - a - - [da 2 + d</< 2 + sinh 2 a.d</> 2 ], 
(5 r) 

where s cosh a, r = cos i/r. Laplace's equation consequently takes the 
form* 

9 /sinh a 9^\ 9 /sinh a 9iA 1 d 2 u __ .. 

^ \ 7 r ^z) "^~ ^./, I o . ^77.) "^"7^ jr^Tr_ 5T2 ~ " (") 



Putting a == v (5 r)i we find that v satisfies the equation 

3 a ^ 82v v X a2v 



in which the variables are separated. Hence there are simple toroidal 
potential functions of type 

u - (s - T)* cos n (0 - ) cosm (^ - c ) [^P m n _ t (5) + BQ m n _^ ()], 
where A, B, n, m, ^> and are arbitrary constants. When TI = | the 
typical solutions may be combined so as to give a solution of type 

u = (5 - r)* cos } (0- ) [/ (< + i x ) + g (0 - t x )], 
where ^ = log (tanh a/2), 

and / and </ are arbitrary functions. This form is indicated by the solution 
of Laplace's equation 



and is also indicated by the fact that 

T (m + 1). P<T m (cosh a) = tanh w (a/2). 

* B. Riemann, Partidle Differentialgleichungen, Hattendorf's ed. (1861); C. Neumann, 
Theorie der ELtktncitat und Wdrme in etnem Hinge, Halle (1864) ; W. M. Hicks, Phil. Trans, vol. CLXXI, 
p. 609 (1881); A. B. Basset, Amer. Journ. of Math. vol. xv, p. 287 (1893); Hydrodynamics, vol. n; 
E. Heine, Anwendungen der Kugelfunktionen, 2nd ed. pp. 283-301, Berlin (1881). 



462 Toroidal Co-ordinates 

We shall now obtain some formulae for the Legendre function of the 
second kind which will be useful in the subsequent work. It should be 
remarked that the more general equation 



a du J. d*u d*u p du ld*u _ - n 

V ^"^ + ^ f ^ a+ 9? ? 3? ?3^ "" ...... 

may be treated by a similar substitution, and it is found that, if 

a + /J 

n + if = m cot \($ + *'"), u = v (cosh a - cos $) 2 , ...... (D) 

then v satisfies the partial differential equation 

d*v ^ dv d*v ., . dv a 2 - 2 

W +aC ih "Wa + W* + P *W~ ~2 * 

, 2 8 2 v 2 . 3 2 i; n 

-f cosech 2 a ^ . 2 cosec 2 y/ ^ 2 = 0. 

This equation evidently possesses elementary solutions of type 

V =S (a) Y(0)O(^)0(0). 
In particular, when a = j3 = 1 we kave solutions of the type 

11=^- r) [4P n (^) + Q n - (5)] [CP n ^ (r) -f JDQ/ (r)] e**+<+, 
where A, B, C and D are arbitrary constants and s = cosh a, r = cos j/r. 
The wave-equation may be reduced to the form (C) by writing 
x = 77 cos <, y = T; sin ^>, 2 = ^ cos 0, ict = ^ sin 0. 

EXAMPLES 
1. Prove that, ifn> 1, p> I, m> 1, ^-fm> l, 

(A,) ./ (Afl) A-*w rfA 

r (p + + 1) F (j> + 1) F ( + 1) 



X ( - r) P- J ' 1>+ ^ B (r) P-Vm-n (), 

where i; and f are defined by equation (D). When m = n this equation reduces to one 
given by H. M. Macdonald, Proc. London Math. Soc. vol. vn, p. 147 (1909). 
2. Prove that 



- r) tanh" (J) tan" (W) - 2a^ J" ^ n (A{) J m (A,) J m+n (Aa) A rfA, 



and deduce that 

*- 

3. Prove that, if m> 0, 



D O n 



4. Prove that P,-(OOB^ ) = 



,- (O08h a) - (4 Sinh a)". 



Associated Legendre Functions 463 

10-2. Jacobi's transformation*. If n is a positive integer we find on 
integrating by parts that 

< (z) fa [(1 - z*)-l]ds. 
Now it follows from the expansion of sin nx in powers of cos x thatf 

d n-l (1 _ z2) -J 1.3... (2n- 1) . , 

(fen-i --- = (-)"" 1 -- ~ -- - am ( cos- 1 z), 

d n 1.3... (2- 1) 

5(1- z 2 )-* = (-)" - vrTp~ cos ( n coa ~ l z >- 

Hence if cf> M (z) is continuous in the range 1 < z < 1, 

[ <f> M (z) (1 - z 2 )"-* dz = 1 .3 ... (2 - 1) f * < () cos (n cos- 1 z) -7== . 

J 1 J-i V 1 2r 

Putting z = cos 6 the equation takes the form 



= 1.3... (2n- 1) (cos 0) cos n# rf0. 
o Jo 

There are many applications of this theorem. If we start with the 
formulae 



r { 

JO 



2 + cos n C08 

O 



we may derive the formulae 

r. / x (n + w) ! (z 2 - I 

m 



cos < 



The last equation is a particular case of the more general equation^ 



which holds when v is not an integer if its real part is greater than 
The first equation may be written in the alternative form 

1 r* 

Z + ~ 



* Crdle's Journal, vol. xv, p. 1; L. Kronecker, Berlin. Sitzungsberichte, S. 539 (1884). 
f An elementary proof of the first equation is given by W. L. Ferrar, Proc. London Math. 
Soc. (2), vol. xxn (1924); Record* of Proceedings, Feb. 14. 
J Whittaker and Watson, Modern Analysis, p. 366. 



464 Toroidal Co-ordinates 

If in Jacobi's formula we take <f> (x) = x m we obtain 



= (m = n + 2s 4- 1) (<$ an integer) 
= (m < n). 
This formula is related to the general formula of Poisson* 

fw/2 

cos" x cos (v + 2m) x dx = 0, 
Jo 

f n '^ 77 1 

cos" x cos vx dx = - . , 

Jo ^ ^ 

where v is any positive quantity and m any positive integer. 

If we next put . a a 2 r 

x (cos 0) --= ( 1 La cos + a*) *, 

< (n) (cos0) = 1.3 ... (2n - 1) a" (1 - 2acos0 + a 2 )~ n ~*, 

Jacobi's formula gives 

fir cosn9.d9 / sin 2w 0.rf0 

I _ _ _^ tt I . ._ 

Jo (1 - 2a cos -f a 2 )* J (1 - 2a cos + a 2 ) n+ *' 

Putting 2s = a -f a~ l and replacing cos in the last integral by t, 
we have 

I (s cos 0)~4 cos nd.dd = 2~ n I 1 - /'-)"-* (,? /)- f -J rf/ 

Another expression for the Legendre function of the second type is 
obtained by making use of the transformation 

cos 9 -= R~^ (cos a), sin </> = jR~J sin 0, 

R = 1 - 2a cos f a 2 , 
(1 a 2 sin 2 </>)~i c/</> = 72~i dO, 

i f" i 

( 1 - 2a cos ^ a 2 )~ n "i sin 2 " . rf0 = ( 1 - a 2 sin 2 ^)~* sin 2w </> . d^. 

. o 'o 

Consequently we have the equations 

TTT eosnO.dO r^ &in 2n <f> . d<f> 

Jo (1 - 2a cos i a 2 )* a J (1 - a 2 sin 2 </>)*' 

p sin 2 " <f>.d<f) _ -n-i/) ri / -i\i 

Jo (1 a 2 sin 2 (f>) n ~* 

* S. D. Poisson, Journ. dc Vticole Polytechnique, Cah. 19, p. 490 (1823). 



Expressions for Legendre Functions 465 

EXAMPLES 

1. Prove that w [2 (* - T)]~* = i e tn * Q w _j (), 

and show that a potential which is constant on the ring s = ,<? is given by 

F = [2 ( - r)]i J jj* Q n ^ (* ) P n _ k W/P n _j K). [Heine.] 

2. Prove that the potential of a uniformly charged circular ring coinciding with the 
fundamental circle z = 0, p = a, is proportional to 

(cosh a cos 0)i sech (%a).K (tanh ja), 
where Til (fc) is the complete elliptic integral with modulus k. 

3. Prove that Laplace's integral for P n {^) can be used to obtain the formula 

JTT.P W (cos 0) = f cos {(n + J) <}. {2 (cos < - cos 0)}~i d<f>. 

[Mehlcr.] 

4. Prove that, when n is a positive integer, 

\n.P n (cos 9) - [" sin {(n + 4) A}.{2 (cos (9 - cos <^)}~4 rf<^. 

7o 

The integrals in Examples 3 and 4 are given in Whittaker and Watson's Modern Analysis, 
p. 315, they have been much used by J. W. Nicholson to evaluate series and integrals 
involving Legendre functions. 

5. Prove that if /(a) is a suitable type of arbitrary function, the differential equation 



is satisfied by the integral 

n+p 

v = / [cos (a 0) cos )3] 2 /(a) da, (s = cos/3). 
7*-j8 

6. Prove that the differential equation 



/^\ 4 , l 2 - 52 - 

W + "^r 



possesses the two particular solutions 

12 - 52 / , 

"- 2 

_ s 3 2 /5\ 3 3 2 .7 2 /5\ 5 3 2 .7 2 .11 2 
Va "" 2 + 31 UJ + ~5T W + ~7l 
the series being convergent for | s \ < 1. 

The corresponding solutions of Legendre's equation of order n may be written respectively 
in the forms 

v 1 = F(- Jn, \n + ; ; s 2 ), v 2 = F (- $n + $,$n + I; $;s 2 ). 

[Heine.] 

10-3. Green's functions for the circular disc and spherical bowl. If R is 
the distance between two points with toroidal co-ordinates (a, /r, </>), 
(tfo> *Ao> </>())> we have 

aR-i = (s- r)l (5 - T )* {2 cosh a - 2 cos (0 - lAo)}"^ --(A) 
where cosh a = cosh a cosh a sinh a sinh a cos (</> </> ). 

B 30 



466 Toroidal Co-ordinates 

The last factor in (A) may be expanded in a cosine series of multiples 
of </r and the coefficient of cos ra (</r ) will be 



cos mifj (cosh a cos 0) * dt/j = (2/7r) Q m _i (cosh a), 
o 
Therefore 

-i = (# r )i ($ TO )J [$i (cosh a) -f 2$i (cosh a) cos (</f </r ) -f ...]. 
This expansion of the inverse distance was given by Heine. 
The series may be summed by writing 

.00 

Q m i (cosh a) = 2~i (cosh u cosh a)~% e~ mu du. 

* Jo. 

This formula is proved in 10-5. The method of summation, which is 
taken from a paper by E. W. Hobson*, leads to the formula 

J a cosh u cos (iff I/JQ) 

In order to obtain the Green's function for a circular disc by an 
extension of the method of images it is convenient to use an idea originated 
and developed by A. Sommerfeldt, and to consider two superposed spaces 
of three dimensions related to one another in much the same way as the 
sheets of a Riemann surface. In the present case the passage from one 
space to the other is made when a point " passes through the disc." The 
two spaces may be distinguished from each other by the inequalities 
TT < if/ < TT in the first space, 
TT < if/ < STT in the second space. 

A point P which starts from a place in the first space on the positive 
side of the disc may pass through the disc into the second space, and ifj 
will increase continuously to a value greater than its original value TT when 
the point is on the disc. In order that this point after the passage through 
the disc may return to its original position P it will be necessary for it to 
pass again through the disc and at the second crossing it returns into the 
first space. 

Corresponding to a point (cr, 0, <f>) in the first space ( TT < if* < TT) there 
is an associated point (or, tff -f 27r, <f>) in the second space. The point 
(a, i/j -f- 47T, <f>) is regarded as identical with the original point in the first 
space. 

We now notice that there is an identity 
2 sinh u __ sinh \u 

COsh U COS (if/ l/r ) "~ COSh \U COS \ (iff ifj Q ) 

sinh \u 
i i * 



COSh \U + COS | (i/f l/r ) ' 
* Camb. Trans, vol. xvra, p. 277 (1899). 

f Math. Ann. Bd. XLVII, Sj 317 (1896); Proc. London Math. Soc. (1), vol. xxvm, p. 396 
(1897). 



Green's Function for a Circular Disc 467 

which indicates that we may write 

R~ l = W (a , to, to) + W (a , fa + ^, fa), 
where 

2<rraW (or ,0 ,</> ) 

0-1 / u / u f sinhjtt.dw , , , i 

= 2 * (5 T )t (s T )* - - __ - (cosh u cosh )~*. 
J a cosh |u - cos I (<A - <Ao) 

Performing the integration we find that 

+ ^n- 1 {cos & (0 - ) sech ( 



It may be shown without much difficulty that this function W is a 
solution of Laplace's equation when considered as a function of either 
a, i/r, (/> or <T O , t/r , (/> ; it is, in fact, a symmetrical function of the two sets 
of co-ordinates. It is, moreover, a uniform function of a, 0, </> in the double 
space since it is unaltered in value when ^ is increased by 4?r. It is con- 
tinuous (D, 1) throughout the double space except at the point (a , > </>o) 
where it becomes infinite like R~ l . It is finite at the point (CT O , 4- 2?r, </> ) 
because at this point sin" 1 [{ ...... }] becomes \TT instead of \rr. It is, 

indeed, the fundamental potential function for the double space. 

Let us now consider the function 

V - W (<r , <Ao> <o) ~ W ((TO, 277 - <A , <), 

which is a potential for the double space when there is a charge at the 
point P with co-ordinates (cr , ^ , </> ) and an image charge at a point P r 
(<T O , 2rr </T O , </> ) which is situated in the second space at the optical image 
of P in the plane of the disc. This function V is infinite in the first space 
only at P and is, moreover, zero on the disc. 

To see this we note that on the disc R is the same for the point P and 
its image, consequently it is only necessary to show that when $ ^ TT 

COS i (iff - </r ) - COS \ (iff - 277 -f </f ), 

and this is evidently true. 

The function V possesses all the characteristics of a Green's function 
and so we may write 

V^G^fa^; oo, fa, fa) = G (Q, P), 

and regard G as the Green's function of the circular disc. It is evidently 
a symmetrical function of the two sets of co-ordinates (a, /s </>), (a , t/r , </> ) ; 
that is, of the points Q and P that have these co-ordinates. 

To solve the corresponding problem for the spherical bowl it is con- 
venient to regard the surface of the bowl as the place where a passage is 
made from one space to the other. If the angle of the bowl is /?, so that 
i/r = ^8 is the equation of the bowl, we must suppose that in the first space 
i/j has values from j8 2ir on the negative side of the bowl up to /? on the 
positive side, and that in the second space iff increases from /J up to /3 + 2n. 
If the convexity of the bowl is upwards, j3 < 77; if it is downwards, j3 > 77. 

30-2 



468 Toroidal Co-ordinates 

We now need the image of P (o- , , <j> ) in the spherical surface of the 
bowl and this must be regarded as a point P' (o- , 2j8 , </> ), which is in 
the second space if j8 - 2rr < < j8. If < < /?, P is above the bowl 
and P' below the bowl. 

The function 



(P, Q) = p + sin- 1 {cot i (0 - ) 

~ -t (cosh CT O cos )^ [cosh CT O cos (2/3 )]^ 

sin^ 1 {cos J (</r + - 2]8) sech } 

[2 TT J 

is seen to satisfy the requirements and is, indeed, the Green's function for 
the spherical bowl. 



[2 



10-4. Relation between toroidal and spheroidal co-ordinates. There is a 
simple relation between toroidal co-ordinates and the spheroidal co- 
ordinates connected with an oblate spheroid. If we write 
p =- a cos cosh 77, z a sin sinh 77, 

, . , 1 ~ COS J/f . , 1 -f COS 

we have sin- =? ; , , sinn- 77 = r . , 

COSh or COS i/r COSI1 cr COS i/f 



cos 





2 



cosh a 1 , cosh a 



j. v^vyojti <- J- 1 o v><v^kjAJL vy i j. 

= ,- ---- r, cosh 2 77 = - , ----,, 
cosh o- - cos ifj ' cosh a cos J/T 

tan | = i sin |/r cosech ^cr, tanh 77 = cos J0 sech |or. 

With the aid of these formulae we may derive the formulae of Lipschitz 
for the Green's functions for the circular disc, and spherical bowl from the 
formulae of Hobson, or vice versa. It is necessary, of course, to pay 
careful attention to the determination of signs where ambiguities are 
produced by the use of the formulae of transformation. 

10' 5. Spherical lens. The potential of an insulated electrified con- 
ducting lens bounded by two intersecting spherical surfaces iff == a ]8 and 
ifj = ^ has been found by H. M. Macdonald*. 

If na = TT, where n is a positive integer, the problem may be solved by 
the method of electrical images. 

We commence by placing a charge K at the centre of the first sphere, 
its magnitude being chosen so as to produce a constant potential nil on 
this sphere. We next introduce a succession of image charges E^ , E 2 , . . . E m , 
chosen so that for E and E l the second sphere is an equipotential for which 
V = 0, for E l and E 2 the first sphere is an equipotential for which F = 0, 
and so on. The second step is similar to the first except that we begin by 
placing a charge E' at the centre of the second sphere, its magnitude being 

* Proc. London Math. Soc. (1), vol. xxvi, p. 156 (1895); vol. xxvm, p. 214 (1896). Refer- 
ences are given in these papers to some earlier work by W. D. Niven. 



Potential of a Conducting Lens 469 

chosen so as to produce a constant potential TrU on this sphere. We then 
introduce a succession of image charges JE/, 2 ', ... such that E' and E 
give a potential which is zero over the first sphere, E and E 2 ' a potential 
which is zero over the second sphere, and so on. 

To express the distance of an arbitrary point from one of the charges 
in toroidal co-ordinates we notice that 

.<? -J- T 

r 2 = o 2 + z 2 - a 2 . 

r S r 

Hence 

(s - r) (r 2 4- a 2 ) = 2a 2 s, (5 - r) (r 2 - a 2 ) - 2a 2 r, (s - r) 2 - a sin 0, 
and so 

2a 2 [5 - cos (0 + 26)] = 2 (s ~ r) [(z sin 6 -f a cos 0) 2 + p 2 sin 2 0]. 

(A') 

Now the z co-ordinates of the different charges are 

E a cot (a - ]3), E' -a cot )3, 

E l a cot a, J57 1 / a cot a, 

#2 a cot (2a - /?), jE 2 ' - a cot (a -f ]8), 

E s a cot 2a, E 3 ' a cot 2a, 



and in each case the co-ordinate is expressed in the form z a cot 0. 
Also the equation (A') shows that the function 

(s - T)* \s - cos (0 + 20)]-* - JF (0), say, 

is a potential which is proportional to the inverse distance from the point 
z = a cot 0. Hence 

F (f$ a) F (a) is a potential which is zero on the sphere = /3, 
F (ft 2a) jF" (a) is a potential which is zero on the sphere = /?, 

and so on. Now JP (j3 a) is a potential which is equal to unity on the 
sphere = a j3, and .F (j3) is a potential which is equal to unity on the 
sphere = ^3. Hence the potential 

V = C/TT [JP (j3 - a) - .F (a) + F (]8 - 2a) - J 7 (2a) -f ... 

4- F () - ^ (- a) + F (ft + a) - J 1 (- 2a) + ...] 

is exactly one which is obtained by the method of images. When n is a 
positive integer we have for ra -f k = n 

F ( - ma) = JF (J8 + fca), ^ (- ma) = jP (fca), 

consequently the charges will repeat themselves unless we take care to 
stop each series at the proper charge. 

The difficulty of knowing when to stop may be avoided by considering 
the potential 

VA = UTT I [F (]8 4- ma) - F (ma)], 
' i 



470 Toroidal Co-ordinates 

which is zero over each sphere. When we put ift = a /? the term F (/? -f- ma) 
is cancelled by the term F [(k -f 1) ], and when </r = /? the term 
F ( + raa) is cancelled by - F (ka), and F (j8 + rca) by - F (na). 

All the terms in the series except F (no) are zero at infinity while 
F (no) -+ 1. We therefore write 

V = F* + t/77, 

and this formula will give us a potential which is zero at infinity and 
constant over the lens. It should be observed that all the charges 
E, E lt E 2 , ... E', EI, E 2 , ... lie within the lens and so our potential is of 
a form suitable for the representation of the potential of the lens. 
Let us now introduce the notation 

f sinh co 

/ W X) 



_ cog y 
Since 

f oo 

(cosh co cosh tr)~i/ (co, x) dco = TT (cosh a cos 

J <T 

we may write 

n //iv If 00 / COS h ff ~ COS \* - . . rt/1 . , 
.F (0) = - - r - ^ / co, /r -f 20) do> 

TT Jo- \cosh co cosh a/ J v r 

n 
Now 



hence we may write 

F = 17 - ^ {/ (n., n^) - / (nc, 



This formula may now be extended to the case in which n is not a 
positive integer. We must first show that F is a solution of Laplace's 
equation and to do this we must show that if 

g (to, ifj) = f (nto, n\lf) - f (nto, mf* + 2nfi), 
the integral 

Too 

v = dco (cosh co cosh a)"* g (co, iff) 

J a 

is a solution of the differential equation 

n d*v ., dv I d*v ^ 

Dv = ^~ + coth ax- + ^v+aT 2 = 0. 

do 2 9cr 4 ci/f 2 

To perform the necessary differentiations we first integrate by parts, 
this gives the equation 

f oo g 

v = 2 dco (cosh co cosh or)i^ (gr cosech co). 

We may now differentiate with respect to a and we find that 
~ = J sinh a (cosh co cosh tr)i ^ (0 cosech co) dco. 



Stream-Function for a Lens 471 

Repeating the process and making use of the fact that g is a solution 
of the equation ^ n ^ 



we find eventually that* 
Dv = da> T x~ cosech 2 co (sinh 2 a sinh 2 co) (cosh co cosh cr)~* 

-f gr cosech 3 co (cosh 2 co f sinh 2 co) (cosh co cosh a) (cosh co cosh cr)i 

= 0. 

A similar result may be obtained with any function g which satisfies 
the equation (B) and behaves in a suitable manner as co -> oo. In 
particular, if g ( ^ ^ = e _ ww cos ^ 

we have v = 2*Q m _i (cosh a) cos miff. 

The stream-function corresponding to V has been found by Greenhillf. 
With the notation qin 



, ir- - 

cosh co cos 

it may be written in the form 

o rr f /COsh CO COSh 

/S = naU \ - r - 
J^ V cosh a- cos ifj 

cosha cos 



77 f/ cosha cos i// \. f r , . ,, . Oxl , 

naU \ [ r ---- r 1 - ) / (co, iff) [h (nco, ?ii/r) h (nco, ^ -f 2n/j)J oco. 
J ff \cosn co cosn o / 



If 8 = (cosh a cos t/i)~i J2, 

the differential equation for R is 



while the relations between V and $ are 

.. ..w ds .. .dv 



f 

) 



When n is a positive integer the expression for S may be verified by 
noticing that if x = <A + 20, 

^ /cosh co cosh <T\ i 
dco -- r - r h(a>, 
V cosh a cos </r / v 

f 00 / cosha- cos ^ \i 
dco r ------ ,~ / (co, ib) h (co, v) 

J a \cosh co - cosh a/ ^ v >r/ v A/ 

= TT cosec [{cos cosh a cos (iff -f 0)} 

x (cosh a cos 0)""* (cosh a cos ^)"1 cos 0], 

* The verification is performed in a slightly different manner by A. G. Greenhill, Proc. Roy. 
Soc. A, vol. xovm, p. 345 (1921); Amer. Jvum. Math. vol. xxxix, p. 335 (1917). The gravi- 
tational attraction of a solid homogeneous spherical lens had been worked out previously by 
G. W. Hill, ibid. vol. xxix, p. 345 (1907) and A. G. Greenhill, vol. xxxm, p. 373 (1911). 

f Loc. cit. 



472 Toroidal Co-ordinates 

while on the other hand 

n cos 6 cosh a cos (iA -f 6) 
=a cosec 8. - cosha _ coaifl - ' 



[< + a cot fl) + ,] _ a cosec . _ 

LV ; ^ J cos 



Hence the foregoing expression is simply proportional to the stream- 
function for a single point charge and so Greenhill's expression may be 
derived from the two series of electrical images. 

An expression for the capacity of the lens is given by 



where S l and S 2 are the values of S at the vertices of the lens. At the 
vertex for which a = 0, </r = j8, ifj -f- 2/3 = j8, we have 

o o rr o/i 0U P (coshco -f l)*doo 

& = 2anl7.8in nfi. (1 - cos 8)* -. -, - - - - m -, /-- -- '\ > 
1 r r/ Jo (cosh to cos j8) (cosh HOJ cos np) 

while at the vertex for which or = 0, i/r = a j8, ^ -f 2)8 = a + )3, we have 
S 2 - 2a7iC7.sinn J 8.[l - cos (a - j3)]* 

f 00 (cosh co -f- 1)4 rfco 

Jo [cosh co cos (a /?)] [cosh nco + cos?ij3]" 

The spherical bowl may be regarded as a particular case of the lens 
in which n = J, a = 27r. The expression obtained for the potential V at 
a point Pis -- 



where 6 is the radius of the sphere = )8, and r x is the distance of P 
from its centre ; furthermore 

siny = 2a/(E l -f ^? 2 )j cosy' = sech |or.cos (|</r 4- j3), 

where JRi and JR 2 are the greatest and least distances of the point P from 
the rim of the bowl. 

The corresponding stream-function is 

S = a7 (1 cos 0)* (cosh cr cos </r)~* 6 (a/ cos 9 l CD), 

where X is the polar angle of the point P when the pole is the centre of the 
sphere i/r = ]8, and the polar axis the line joining the centres of the two 
spheres. This result is due to J. R. Wilton* and A. G. Greenhillf. 

10-6. The Green's function for a wedge. If we invert the lens from an 
arbitrary point we may obtain the Green's function for the inverse lens. 
If, however, the point is on the rim of the lens the surfaces of the lens will 
invert into planes and we shall obtain the Green's function for a wedge 
formed from two semi-infinite conducting planes which intersect at an 

* Mesa, of Moth. (1914). f Loc. cit. ante, p. 471. 



Green's Function for a Wedge 473 

angle ir/n. This problem was solved by a direct method of A. Sommerfeld* 
and H. M. Macdonaldf. 

If (/>, z, (f)) are cylindrical co-ordinates, </> = 0, </> = TT/H, the equations 
of the conducting planes, and if an electric charge is placed at the point 
(//, z', </>'), the potential F is given by the formula 



dt, (cosh - cosh -n)^[f(n^n(f>-n^)-f(n^ 

where 

2pp' cosh 77 - P * + p' 2 + (2 - z') 2 . 

The potential can also be expanded in the form of a Fourier series 



V = 4ne S F mn (77) sin (wn</>) sin (ran<//), 

m-l 

fo) = (2p,/r* r^ (cosh - cosh 7,)-* e- mnc 



m-l 

where 



An alternative expression for F mn is 

*-*'\J mn (K P ) J mn ( K p f ) d K . 



o 

This may be deduced from a well-known expression for the Q-f unction J 
or may be obtained directly. 

When n = m + J, where m is a positive integer, the potential can be 
expressed as a finite sum, for then, if (2m -f 1) a = 2?!, 

2m foo 

= e (2pp')-i S d (cosh g - cosh r,)~i [/ (K, ^ - W + sa) 

s=0 J >j 



* - * 

-f ' + 2,a tan- 



where for brevity we have used the notation 

(a, 6) = (cosh a cos 6)~i, 
[a, 6] = [cosh a -f cos 



10-7. TAe Green's function for a semi-infinite plane. When m = this 
expression gives the potential when a point charge is in the presence of 

* Proc. London Math. Soc. (1), vol. xxvin, p. 395 (1897). 

t Ibid. vol. xxvi, p. 156 (1895). 

J H. M. Macdonald, Proc. London Math. Soc. (2), vol. VIT, p. 142 (1909). 



474 Toroidal Co-ordinates 

a conductor in the form of an infinite half plane. Since a = %TT the potential 
is simply 



10-8. Circular disc in any field of force. H. M. Macdonald* has con- 
sidered the case in which the potential of the inducing system can be 
expressed in the form 

S S A nv J n (vp) cos (n</> + a,), 

where A nv , n, v and a,, are constants. The solution of the simplified 
problem in which the series reduces to a single term was given by Gallop 
for the case n = 0, and by Basset for the case n = 1 . We shall commence 
the discussion of the general case by considering the formulae 

Too 

(s - r)* cos %ifj . tanh (a/2) = a* yV e ~ KZ J ( K p) ^m-i ( Ka ) 

J o 

w> - |, z> 0, 

Too 

(s r)* sin i/r.tanh m (or/2) = a* V 77 - e~* z ^m (KP) J m+ $ (/ca) ^d/c 

'o 

m > 1, ^ > 0. 

Each expression multiplied by cos m<f> represents a solution of Laplace's 
equation and so each expression divided by p m is a solution of the equation 

d*u 2m + 1 &w 2 2 u _ 

ap + p a^ + a?*" 

Since the solution of this equation is determined uniquely by its value 
on the axis of z it is sufficient to verify the truth of the formulae by making 
p -+ after- dividing by p m . The integrals are uniformly convergent in the 
neighbourhood of p after this operation and so we have merely to 
verify the equations 



TOO 

(m + l).cos J0. (1 - cos 0) w+ * = aS +m vV e- 

J 



'0 

These are easily seen to be true on account of Sonine's formulae 

(t) r+* dt = T (m + 1) 2 w+ i x (1 + x*)- m ~ l (m> - |), 

(t) r+i dt = T (m + 1) 2 W+ * (1 -f ^ 2 )- m ~ 1 (m > - 1). 
* Proc. London Math. Soc. (1), vol. xxvi, p. 257 (1895). 



Disc in a Field of Force 475 

It may be observed also that the value of the second integral may be 
deduced from that of the first with the aid of the substitution 

(cosh or cos i/f) (cosh <j cos x) = sinh 2 o-, 
which gives _ p sinh a _ p sin x 

cosh o- cos x ' cosh a cos x ' 

cos fyfi (cosh a cos $)% = sin ^.sinh a (cosh o- -f l)i (cosh o- cos x)~ l > 
sin J(/r (cosh a cos ^) = cos ^.sinh o- (cosh a l)i (cosh cr cos x)" 1 - 
The value of the first integral for p > a can now be deduced from the 
value of the second integral for p < a by interchanging p and a and writing 
m % instead of m. 

Let us now consider the potential 

V = (s r)i cos ^i/j. jbanh m |o-.cos m</. 
When z = and p 2 < a 2 we have = TT, consequently 

F - 0, a -K- = 2* cosh 3 ( Ja) . tanh w ($a) . cos w<. 

When 2=0 and p 2 > a 2 we have i/j = 0, consequently 

37 A 
F = 2* cosh |a. tanh m ( Ja) . cos m</>, g~ = u - 

Next, consider the potential F = W m cos m^, where 

W w = f^e-^rfic f J m _j (vo) t/ m _ t (ico) / w (icp) M* ada. 
Jo Jo 

Integrating under the integral sign the first formula tells us that when 
2=0 and p < c, 

W m = V(2^/7r) f e/ m _j M p- (/> 2 - a 2 )-* a+* da 
Jo 

r i7r 

I ^m- (vp sin 9) . sin m+ i . d6 



Again, when 2=0 and p < c, 
dPF f f c 

~ = - rf/C V(^) J m- 

C 1 ^ JQ Jo 

= - V J m (pv) + r dK\ 
JO Jc 

= - vJ m (pv) + V(^M jP m (a 2 - p 2 )-* JV 
When 2=0 and / c, we have on the other hand 

,sin- l (c/p) 



,s 

77) 
Jo 



sn 



* 6 . d, 



CHAPTER XI 
DIFFRACTION PROBLEMS 

11-1. Diffraction by a half plane. The problem of diffraction of plane 
waves by a straight edge parallel to the wave fronts or surfaces of constant 
phase was shown by Sommerfeld* to be one which could be treated 
successfully by the methods of exact analysis. The analysis has subse- 
quently been expressed in different forms, and various attempts have been 
made to make the derivation of the final formulae seem natural and 
straightforward. It must be confessed, however, that the discovery of any 
of these methods requires remarkable insight and a very thorough know- 
ledge of the different types of solutions of the wave-equation, and the 
solution of the diffraction problem must be regarded as a triumph of 
mathematical ingenuity and experiment. 

The method which will be followed here is one which was devised when 
the solution was known ; it has the advantage that it presents the solution 
in a particularly simple form. 

If F (x, y, z, t) is a solution of the wave-equation D 2 F = it may be 
easily verified that the definite integral 

f7r/2 r r l 

V = I Fix tan 2 , y tan 2 a, z tan 2 a, t sec 2 a tan a da 

Jo I c J 

is a solution of the wave-equation provided that 

a 
di 



F x tan 2 a, y tan 2 a, z tan 2 a, t sec 2 a \ tan 2 a 

has the same value when a = ^ as it has when a = 0, and that the function 



F behaves in a suitable manner for values of its arguments which are 
either zero or infinite. 

Again, it is easily verified that the function 

F (x, y, z, *)=(* + y)~* (p, *, t) 

is a solution of the wave-equation if G (x, y, t) is a solution of the two- 
dimensional wave-equation 



~dx* dy* ~ c 2 W ...... 

* Math. Ann. Bd. XLVII, S. 317 (1896); Zeits. fur Math, und Phys. Bd. XLVI, S. 11 (1901). 
See also H. S. Carslaw, Proc. London Math. Soc. (1), vol. xxx, p. 121 (1899); Proc. Edinburgh Math. 
Soc. vol. xix, p. 71 (1901). An approximate solution was given by H. Poincare, Acta Mathe- 
matica, vol. xvi, p. 297 (1892); vol. xx, p. 313 (1896). 



Solutions of the Wave-Equation 477 

When the first theorem can be applied to this function, it tells us that 
the expression 

fT/2 r r -I 

(x -f iy)~* G \p tan 2 , z tan 2 a, t sec 2 a da 

Jo L c J 

represents a solution of the wave-equation, and since it is of the form t 

(x + iy)~lH (p, z, t) 
the natural inference is that the integral 

G\x tan 2 , y tan 2 a, t ^ sec 2 a rf 

Jo L c J 

must be a solution of (A). The foregoing method of deriving one wave- 
function from another seenLs to be applicable, then, to wave-functions in 
a space of any number of dimensions. 

Let us now consider a diffraction problem in which the primary waves 
are specified in some way by means of the wave-function 
cf) = (f) Q = F (ct -f y sin -f x cos ), 

where < may be the velocity potential of waves of sound or one of the 
electromagnetic vectors if we are dealing with waves of light. Tf these 
waves are reflected completely at the plane y = there is a reflected wave 
specified by the wave-function 

^ = fa = F (ct -f x cos OQ y sin ), 

and the complete wave-function is </> = (/> -f </>i when the boundary con- 
dition is x- = for t/ = 0, but is </> = </> fa when the boundary con- 
dition is (f> = for y = 0. 

When the primary waves are diffracted by a screen, ?/ 0, a; > 0, 
which occupies only half of the plane y = 0, it is necessary to divide the 
xy-plane up into three regions S lt /S 2 , $ 3 , whose boundaries are as follows : 

$1 ? y = 0, a; sin -f- iy cos = ; 
S 2 , # sin OQ y cos = ; 

<S 3 , y = ' ^ sin ^o - y cos - o. 

The limits of 8^ are thus the screen and the geometrical limit of the 
reflected wave, the Limits of $ 2 are the geometrical limits of the reflected 
wave and the shadow, the limits of S 3 are the screen and the geometrical 
boundary of the shadow. In each case the geometrical limit is obtained by 
the methods of geometrical optics. 

To take into consideration the phenomena of diffraction we associate 
with fa a wave-function F defined by the equation ' 

i r*/ 2 

V = - F [ct -f x cos + y sin (p + x cos -f- y sin ) sec 2 a] da, 

f Jo 

t The method ia an extension of one used by F. J. W. Whipple, Phil.JfoO; (^). vol. 30, p. 420 
(1918) and by W. G. Bickley, ibid., (6) vol. 39, p. 668 (1920). 



478 Diffraction Problems 

and we associate with fa a wave-function V t denned by the equation 

1 ("I* 
V = F [ct + x cos 6 Q y sin (p -f x cos - y sin ) sec 2 a] da. 

TT JO 

We shall try to prove that the conditions imposed by the two boundary 
problems may be satisfied by using a complete wave-function (f> which is 
defined in the different regions as follows: 

< = <o + </>!- F - F! in #!, 



n 
in 



We have to show that this function </ and its first derivatives are 
continuous as a point passes from S t to S 2 , and as a point passes from 
/Sj to S 3 . Now, when x = p cos , y = p sin 6 Q , we have 



- _ 

o- <>o, aa . - 93, 9y " 2 dy 9 
and when x = p cos , y = /o sin , we have 

F __:u 9Z_i_ ia fi aj^ 1 ^. 

x ~ t9lJ 9x 2 ~dx ' dy 2 3y ' 

hence the requirement of continuity is seen to be satisfied. The boundary 
condition is also satisfied on both faces of the screen and <f> is continuous 
(Z), 1) over the whole plane, when the screen is regarded as a cut, hence 
it will give the solution of the boundary problem if it has the correct form 
at infinity. 

This is certainly the case if F (s) is zero when | s \ is greater than some 
fixed number, that is, if the incident wave is limited at the front and rear, 
for then the form of </> at infinity is precisely that given by the methods 
of geometrical optics, F and V l being zero. 

The interesting case of periodic waves may be discussed by writing 
F (s) = e lks . The expressions for </> may then be reduced to those of 
Sommerfeld by writing 



1 f 7 '/ 2 /?\ f-'sl 

1 g-^sec'a^^ (*y e-^dr. 

7TJQ W/ J-oo 

This identity may be derived by writing the right-hand side in the form 

2i f-Ul r 

~ dr c-'^+^da, 

fr J-oo Jo 

a transformation which is possible on account of the well-known equation 



The repeated integral is now replaced by a double integral and trans- 



Sommerfeld's Integrals 479 

formed by means of the substitution r = z cos a, a = z sin a. Integrating 
with respect to r we obtain 

1 f/ 2 

e- wasec * a da. * 

^Jo 

This supplies the outlines of a proof. 

It may be remarked that in the present case it cannot be proved by 
direct differentiation that the integrals occurring in the solution are wave- 
functions. A transformation is therefore advantageous. 

11-2. The various steps sketched above may be justified without 
much difficulty, but it is more interesting to examine the steps by which 
Sommerfeld was led to his famous solu- 
tion of the problem*. Let 

In (z) = log [e*/" - <*/], 

where n is any positive integer, then, \i U S U S 

C is a small circle enclosing the point 
z = # in the z-plane and no other sin- 
gularity of the integrand, the contour 
integral 



// (z) dz 



represents the function 



I 



e 



U 



S 



U 



Fig. 29. 
S, shaded region; U, unshaded region. 



which is known to be a solution of the 
equation V% -h k*u = 0. Our representa- 
tion of this solution still holds when 
the circle^ G is replaced by a path C (0) 
consisting of two branches which go 
to infinity within the shaded strips of 
breadth TT in which e ifcpcos( *- e o> has a 
negative real part. These two branches may, in fact, be joined by dotted 
lines, as indicated in the figure, so as to form a closed contour which 
can be deformed into the circle C without passing over any singularity 
of the integrand. The integrals along these dotted lines, moreover, cancel 
on account of the periodicity of the integrand and so the integral round G 
is equal to the integral round the path G (6) consisting of the two branches. 
The function 



is also a solution of V 2 ^ + k*u = 0, since the necessary differentiations can 

* Our presentation in 11-2-114 follows closely that given by Wolfsohn in Handbuch der 
Physik, Bd. xx, pp. 2G&-277. 



480 Diffraction Problems 

be performed under the integral sign. Its period, however, is 2n77, and so 
to make it uniform we must consider a Riemann surface R with n sheets. 
When 9 incf-eases by 2^77 the path C (9) is displaced a distance 2ri77 to the 
right but, since the integrand has the period 2^77, it runs over the same set 
of values as before. 

This function u n has the following properties : 

If | 9 - | < a < 77, u n -> u as p -> oo. 

If TT < /? < | |, u n -> as p -> oo. 

Indeed, if | | < a < 77, we may deform the path of integration so 
that each of the new branches runs in shaded regions only. In the first 
sheet of R we enclose the point 9 Q by the circle C as before. 

As p -+ oo the integral over the path in the shaded region tends to 
zero. In the first sheet of R the integral round G gives u ly therefore 

? y __ V (1) I 7 y (2) I 7 y (n) 

**! ~~ a n ^ u n i u n > 

where u n (s} is the value of u n at the point lying over />, 9 in the sth sheet 
of R.. If denotes the integral over the contour C (9 + 2,577), we have 
in fact 



/I J2 

where F denotes the path made up of the branches 

C (0), C (9 -f 277), ... C[0+2(n-l) TT]. 

Converting this into a closed contour by broken lines as before and noting 
that the integrals along the broken lines cancel, we have a contour which 
can be deformed into (7, and so the result follows. The proof for the other 
case follows similar lines except that now the contour may be reduced to 
a point by a suitable deformation. 
We now put n ~ 2 and write 



f sin J-Jo,T^ 
JC(B) 2 



JC(B) 
The path of integration can be deformed into the lines 

z == 9 77 + ir and z = -f TT -f ir oo < r < oo, 
after which the integrations can be performed, giving 

) -COB e-*** V - a G (T), 



u, 



where a^-^rr, O(r) = e-^dr, !T = 
V i?r Jo 



Waves from a Line Source 481 

Since u 2 (1} -f u z (2} = ^, we may write 

t^ci) = |^ {1 -f- aO(r)} < < 277, 
^ 2 (2) = ^ {i + a G(- T )} - 277 < 8 < 0. 

Also, since aG ( oo) = 1, it is easily seen that 

i^u) = u 2> o < 9 < 277; ^ 2 < 2) = u 2 , - 277 < < 0, 

f T 

where w 2 = % . e iff / 4 . 77 * e~ tr * dr. 



This is Sommerf eld's expression. When this function w 2 is multiplied 
by e lkct a wave-function v = u 2 (/>, 0, ) e lfcci is obtained. If 

JP [p, 0, *, , T] = [c (* - r) + P cos (6 - )], F = e^ FT-1, 
this wave-function can be written in the form 



v = - 



(t"/7r)i f Tl Fdr, cos J (0 - ) < 

J 00 

- - \ck (i/7r)* { f Tl Fdr + 2 f T ' Fdrl , cos \ (0 - ) > 0, 

(J-oo J Tl ) 

T! = t - p/c, T 2 =t + ?cos(e- 6> ), 

L> 

where the integrand is in each case a wave-function for all values of the 
parameter. It should be noticed that r 2 is a value of T for which 

W[p 9 9 t 9 Q , r]=0. 

11-3. The chief advantage of the last expression for v is that it 
suggests the form of the solution for the case in which the waves originate 
from a line source (p 0) # ) which may be either at rest or in motion. 

Let us take as the potential of our line source the function 

f T/ e ickr dr f T ' 

<f>Q= . ------------ ---------------------------------- - = F (r) dr, say, 

J _oo [ C 2 (t _ r) 2 _ p 2 _ ^2 + 2pp Q COS (6 - fi )] J-oo 

where T' is a value of r less than t, for which the denominator of the 
integrand is zero. When the line source is in motion p and are functions 
of T, we suppose them to be functions such that the velocity of the line 
source is always less than c. For the reflected wave we write 

/ f T " e tkcr dr f T// ~, x , 

0. = ---- . - ^= O (r) rfr, say, 

J _> [C 2 ( ^ _ r)2 _ p2 _ po 2 + 2pPto cos (9 -f )] J J 

where T" is a value of r less than , for which the denominator of the 
integrand vanishes. These integrals are generally wave-functions. 
Now let TJ be a value of r defined by the equation* 

TI = t - - [p + po (TJ)], 
t/ 

* Since p a + p^ - 2pp cos (^ ) < (p + p ) 2 , it is evident that (t - rtf is greater than either 
(t - r') 2 or (t - T r/ ) a , consequently r 2 < r and r x < T". 

B 31 



482 Diffraction Problems 

then, if the boundary condition at the surface of the screen is J- = 0, 
the solution of the diffraction problem may be expressed in the form 

l F (T) dr + [ T F (T) dr -f | P G (r) dr -{- [ G (r) dr in S l9 

oo J T! J oo Jr t 

J^ ( T ) dr + ( T JF (T) dr + P (7 (T) dr in S 2 , 

CO J T l J 00 

^ = | [ n jF ( T ) d T + I [ Tl (? ( T ) dr in 3 . 

J 00 J 00 

The integrals with the limit r x being also wave-functions, as may be 
verified by differentiation. 

The space 8^ is bounded by the screen and the limiting surface of the 
geometrical shadow for the image of the source, the space S 2 is bounded 
by the limiting surfaces of the geometrical shadows for the source and its 
image, while 8 3 is the space bounded by the screen and the limiting surface 
of the geometrical shadow for the source. 

The boundary of the geometrical shadow for the source is defined by 
the equation r x = T', while the boundary of the geometrical shadow for the 
image of *the source is defined by the equation r = T". It is evident, then, 
that (f> is continuous at the boundaries of the shadows. That the first 
derivatives of </> are also continuous may be seen by transforming the 
integrals, in which F (r) is the integrand, by the substitution 
c 2 (t - r) 2 = [/> 2 + /> 2 - 2 PPo cos (6 - )] cosh 2 u, 
arid the integrals in which G (r) is the integrand, by the substitution 
c 2 (t - r) 2 - tp* -f- A, 2 - 2pfr cos (9 -F- )] cosh 2 v. 

In the case when p Q and are constants and the line source is con- 
sequently at rest, these substitutions transform the expression for </> to 
Macdonald's form* 

r TWO rvo n 

^ = ^e ikct e~ ikRc Bhu du 4- e -ikR' COB* v fi v \ ? 

L J <x> J oo J 

where 

R sinh u - 2 (/^i cos \ (6 - ), JB' sinh v = 2 (/>/> )* cos i (0 + ), 
J? 2 = P 2 + Po 2 ~ %> cos (0 - ), J?' 2 = p 2 -f /> 2 - 2^ cos (0 + ). 
It should be noticed that when we transform from u to r in the first 
integral the path of integration runs from r = oo to r = TJ if u is 
negative, but when u is positive the path runs from oo up to a maximum 
value T = T' and then back to r x . A similar phenomenon occurs in the 
transformation of the second integral. This accounts for the existence of 
three different expressions for <j> when r is taken as the variable of 
integration. 

* Proc. London Math. Soc. (2), vol. xiv, p. 410 (1915). 



Waves from a Moving Source 483 

Let us next consider the case in which the rectilinear source moves 
with constant velocity along a path which does not intersect the screen 
and is such that the co-ordinates of a point at time r are 

*o (T) = 4- ar, y Q ( T ) = 17 + br, 
where and 77 are constants and a 2 -f b 2 < c 2 . In this case we may write 

(c 2 - a 2 - ft 2 ) T - cH - a (x - f) - b (y - T?) - 8 cosh u, 

(c 2 - a 2 - b 2 ) T - cH - a (x - g) + b (y + >?) - 5' cosh v, 
where 

a (a - ) - 6 (y - 7?)} 2 - {c 2 - a 2 - 6 2 } 



' - a (x - g) -h b (y + T?)} 2 - {c 2 - a 2 - b 2 } 

x {c 2 t 2 (x ) 2 (y + ??) 2 }]^; 
then 

., a (a; -) -f 6 (y - 77) +S cosh it 



-fje"** 8 ^r^ZftS dvly 

J-oo ' J 

where 
S cosh w = /oc ax by 

+ [{c (ct - p) + a -f 6^} 2 ~ {(^ - p) 2 - 2 - T? 2 } {c 2 - a 2 - 
$' cosh v Q = pc ax -}- by 

+ [{c (c* - />) + a| + 6^} 2 - {(c* - p) 2 - ^ - >7 2 } {c 2 - a 2 - 6 2 }]i 

11*4. Discussion of Sommerf eld's solution. For T < we obtain by 
partial integration 



Therefore 



l 



Introducing a quantity to indicate the precision with which measure- 
ments may be made, we have for points outside the parabola ^-j r- < c, 

the approximation formula 

I 



"' - 2iT' 
Similarly, when T > 0, 



T<0. 



31-2 



484 Diffraction Problems 

These results may be used to obtain approximate representations of 
the light vector when plane waves of light are diffracted at a straight 

edge. If 

i f^i 

Vl = ikp COS (0 - V + t (ir/4+fccO ^-i g->' fir , 

J oo 

where T : = V2kp cos ^-^ , 



and v 



[ T * 
77~i e~ lT * dr, 

J 00 



where _ y cos 



2 ~- ' 

the solution of a number of diffraction problems may be expressed in 
terms of v t and v 2 . In particular, if plane waves of light are incident upon 
a screen (y = 0, x > 0) which is a perfect conductor, and the waves are 
polarized so that the electric vector is parallel to the edge of the screen, 
the electric vector E z in the total electromagnetic field is given by the 
expression E = R{VI _ V ^ ...... (B ) 

where the symbol R is used to denote the real part of the expression which 
follows it. This expression evidently satisfies the boundary condition 
E z = when y = 0, x > 0, i.e. when 9 = and 6 = ZTT. On the other hand, 
if the magnetic vector is parallel to the edge of the screen, the magnetic 
vector in the total field is given by the equation 

#, = fl fo + t> 2 }. ...... (C) 

37^ 
The boundary condition which is satisfied in this case is -~-* = 0, which, 

^ , 1 * i j ,. dHz 3H y ldE x . - i A A 

on account of the held equation ^ ---- ^ ~1/ > 1S e( l m valent to 

~ Iz =r or 8 ^ z = 0, when 6 = and - 2n. 
dy 



Let 8 = ^- VWp) cos (kp - kct + ^ J , 



then in the region of the geometrical shadow (?\ < 0, T 2 < 0) we have 
respectively in the two cases just considered 

^ o / -I- 9 - 9 Q \ / + 9 Q - Q \ 

E z - S (^sec - - sec -y 2 ] > H z = - S \sec y- + sec J . 

In the region Tj > 0, T 2 < which contains the incident but not the 
reflected wave, we have the approximate expressions 

E z = cos {kp cos (9 - ) + kct} + S (sec ^^ - sec 
IT, = cos {kp cos ( - ) + kct} - S sec ^ + 



sec ^ 






Discussion of Sommerfeld? s Solution 485 

and in the region T t > 0, T 2 > 0, which contains the reflected wave, we 
have approximately 

E z = cos {kp cos (0 ) 4- kct} cos {kp cos (0 4- ) 4- kct} 

+ 8 L ec 6 + o_ sec - e 

T~ *J ^OC^ "T C5CVy ~ 



H z = cos {kp cos (0 - ) + kct} + cos {kp cos (0 4- ) + fccQ 

-f- - ) 





The disturbance diverging from the edge of the screen like a cylindrical 
wave whose intensity falls off like p~i is called the diffraction wave. The 
phenomena of diffraction may be regarded as the result of an interference 
between this wave and the incident and reflected waves. 

There is no true source of light at the edge of the screen, yet the eye, 
when it accommodates itself so as to view the edge of the screen, receives 
the impression that the edge is luminous. Gouy and Wien first observed 
this phenomenon in the region of the geometrical shadow where the 
phenomenon is not masked by the incident light. 

For the amplitude of the electric vector in the diffracted wave we have 
in the two cases 



A, - v0 sec 



respectively, where in the second case A 2 is the amplitude of the electric 
vector perpendicular to the edge of the screen, and where in the first case 
A l is the amplitude of the electric vector parallel to the edge of the screen. 

If the incident waves are linearly polarized so that they can be resolved 
into a wave of amplitude % with electric vector parallel to the screen, and 
a wave of amplitude a 2 with electric vector perpendicular to the screen, 
the amplitudes of the corresponding components of the diffracted wave 
are respectively a 1 A l and a 2 A 2 . Since these are no longer in the ratio 
a x : a 2 there is a rotation of the plane of polarization. 

When the screen is illuminated with natural light in which waves with 
all phases and directions of polarization occur with equal frequency we 
have a x = a 2 , but since a^A^ =f #2^2 the diffraction wave is polarized. 

It should be noticed that 





For the case of perpendicular incidence = ^ , we have 



where 8 is the angle of diffraction. 



486 Diffraction Problems 

The measurements of W. Wien* with very fine sharp steel blades show 
a somewhat stronger increase of this ratio than that indicated by this 
formula. Epsteinf attributes this to the finite thickness of the blade. 

Raman and KrishnanJ have recently invented a new method of dis- 
cussing the influence of the material of the screen in which the solutions 
taking the place of (B) and (C) are respectively 



where /? and y are complex constants depending on the nature of the 
material and the angle of incidence ifj = -= # . This amounts to a change 

2t 

in the boundary condition, the solutions adopted being still wave-functions. 
If o> denotes the material constant n (1 i/c), where n is the refractive 
index and K the index of Absorption, the values adopted for /J and y are 
respectively 

Q __ a ~ COS *f* _ 6t)2 COS ~~ a 

P ~~ a + cos iff' ^ ~~ o> 2 cos if/ + a ' 
where a 2 = a> 2 sin 2 if*. 

In this way an explanation is obtained of the results of Gouy relating 
to the effect of the material on the colour and polarization of the diffracted 
light. 

11-5. Use of parabolic co-ordinates. It was shown by Lamb and 
later by Epstein || and Crudeli^f that the problems of diffraction can be 
treated successfully with the aid of the parabolic co-ordinates , T? defined 
by the equation x + iy = ( ^ + ^ )2> ^ > Q 

In terms of these variables we have 

%<$> d<(> 
- 



A particular solution of the wave-equation 

**+_ 

dt* 

is r~*cos \0.f(ct - r), 

where/ (ct r) is an arbitrary function with continuous second derivative. 

* Wied. Ann. Bd. xxvm, S. 117 (1886). 

t Diss. Munich (1914) ;' Encyklopddie der Math. Wise. Bd. v. 3, Heft 3 (1916). 

J C. V. Raman and K. S. Krishnan, Proc. Roy. Soc. London, A, vol. cxvi, p. 254 (1927). 

H. Lamb, Proc. London Math. Soc. (2), vol. vin, p. 422 (1910). 

|| P. S. Epstein, Diss. Munich (1914). 

If U. Oudeli, 11 Nuovo Cimento (6), vol. xi, p. 277 (1916). 



Parabolic Co-ordinates 487 

T^fk 
Let us denote this function by - , then the equation for < is 



and a solution which satisfies the condition ^ = 0, or -^ = at the 

dy 877 

screen rj = 0, is 

V - cr 2 ) dor + J *f(ct-y- a 2 ) da 



where .F is another arbitrary function. Now when is nearly equal to TT 
and r is very great, the upper limits of the integrals become oo and oo, 
respectively, and the asymptotic form of <f> is 



-f y a 2 ) da - f/ (ct - y - a 2 ) 



this is identical with F (ct 4 y) if 

[*/ (y - a 2 ) da = JJF (y), - oo < y < oo. 
Jo 

This is an integral equation for the determination of the function / 
when F is given. Writing a = (y v)* the equation takes the form 



rV 

\ (y-v)-*f(v)dv, -oc<i/<oo, 



and the solution given by Abel's inversion formula is 



If lim F (u) = and lim [(v - u)% F (u)] exists, 

M-> 00 M->~00 

/ () = i J* Jim [(t, - )* JT ()] + M (v- )-* J" (u) d. 

77 ttt? w _^ -.QO 77" J oo 

In particular, if F (u) = ( ^)* (u), where G (- oo) has a finite value 
different from zero, 

lim [(v - tO* J? 1 (w)] = (- oo), 

ti-> oo 

and we have simply 

1 f v 
/ ( V ) = - (t? - w)-t J?" (w) efo. ...... (D) 

""" J-OO 

The foregoing conditions imposed on F (u) are not necessary for the 
existence of a solution given by the formula (D), for if F (u) = cos ku, 
we have F' (u) = k sin ku, 

k ( v i 2k f 

-- (v u)~*sin kudu = -- sin {k (v s 2 )} cfo 

7T J-oo 77 Jo 

= (t/w)* cos (At; + 77/4), 



488 Diffraction Problems 

and it may be verified directly that / (v) = (&/TT)* cos (kv + 7r/4) is a 
solution of the integral equation 

dv. 



r 

= 

J 



11-6. In the work of Epstein an endeavour is made to allow for the 
influence of the material of which the screen is made, and so the surface 
of the screen is taken to be a parabolic cylinder and the material is supposed 
to have a finite conductivity. The electromagnetic field vectors E and H 
are regarded as the real parts of the expressions eT, hT, respectively, 
where T denotes the exponential factor exp (int). In a material medium 
with constants /c, 0, p and a the equations satisfied by the vectors e and 

h are curl A = ae, curl e = - |8A, 

div e = 0, div h = 0, 

where inn a n ian 

a = - - + - , p = - . 
c c r c 

These equations are transformed by the substitution 

x = \ ( 2 - ^ 2 ), y = ?, * = z> s 2 = 2 + 7?', 
into equations connecting the new components e^ , e, , e z , /^ , A, , h z , 

3 9 - 3A Z 

( ^ ~ ( rf ^ a6 * ' 



These equations imply that e z and h z are solutions of the partial 
differential equation 



where P = - ajS 

C 

The boundary conditions at the surface of the screen are, when /u, = 1, 



where F = 6 Z or h z , according as the incident wave is polarized perpendicular 
to the edge of the screen or parallel to the focal axis of the cylinder, and 
where y is unity in the first case and k~ 2 in the second. 

The differential equation for V may be satisfied by writing 

V-X&YW, 

where X and Y satisfy the differential equations of Weber * 



where A is an arbitrary constant. Writing A = 2ik (n -f J), 
X = exp (- ik*/2),U n [f ^/(ik)l Y - exp (ik^/2) U 
* H. Weber, Math. Ann. Bd. i, S. 1 (1869). 



Conducting Parabolic Cylinder 489 

we have particular solutions expressed in terms of the polynomials of 
Hermite 

n\ X n (f) = 

n\ Y n (rj) = 

These are suitable for the representation of an electromagnetic dis- 
turbance in the interior of the parabolic cylinder. Distinguishing the 
functions associated with the value of k for the interior of the cylinder by 
a star, we have for the interior of the cylinder 

F* = S b n X n * (f) Y n * (r,). 
n=0 

To represent the disturbance outside the cylinder it is necessary to use 
the second solution of the differential equation, and this may be chosen so 
that for 77 > n there is an asymptotic representation 



where Z n (77) denotes this second solution. We may now assume for the 
space outside the cylinder 

+ i a n X n () Z n (7), 



n=0 

= sec (0 ) S n ! tan 11 (J0 ) X n (f ) 7 n (77), 

n-O 

where the first term represents the incident wave and the coefficients a n , 
like the coefficients 6 n , are to be determined by means of the boundary 
conditions. The analysis has been formally completed by Epstein and some 
rough calculations made. 

In the case of the infinitely thin parabolic cylinder (or half plane) 
which is a perfect conductor an agreement is found with the formula of 
Sommerfeld. When the thickness is not quite negligible and the conduction 
is still perfect, it is concluded that the term representing the correction 
will increase the amplitude in the case || and decrease it in the case J_, 

so that the ratio A 2 /A l is greater than the value tan f |8 -f -j found 

in 11-4. This is in qualitative agreement with observation. With finite 
conductivity the parabolic screen gives a selective effect favouring wave- 
lengths which are most strongly reflected by a plane mirror of the same 
material. This selective effect is extremely weak in the case J_ and quite 
appreciable in the case 1 1 . These predictions of theory have been confirmed 
by recent experiments*. 

* F. Jentech, Ann. d. Phya. Bd. LXXXIV, S. 292 (1927-28). 



490 Diffraction Problems 

EXAMPLES 

1. Prove that if f(x) = j e x ~ r V 

] x r 

the definite integral F = e ci ~ z [* 2 e -< f + 2) ton2a tan ada 

y o 

represents the wave-function V = %e ct ~ r f(r -h z). 

2. Prove that if n > 1, 

/"*' V tan 2 a t^n ada r (5-+- 1 ) T e 

and write down an expression for a solution of 

a 2 F a 2 F a 2 F i a 2 F 

K o T~ ^ i> T V 9 o 3*2 

dx^ dx n 2 oz 2 c 2 ot 2 

which is of the form F = e ct ~ r f(r + z), 

where r 2 = ^ 2 -h ... z n 2 -f 2 2 . 

3. Prove that if 

ct + y b tan a, ct y ^= b tan ]3, ct r = b tan w, ^ < (a, ft w) < ^ 
the wave-potential 

^ ^ 2ft ^ C S2 a ~^ C &2 & + ( + V) b~^ COsi co COS a COS (Jo) -f a + i^) 

+ (f ->?)&"* COsi cu COS COS (Ja> -f j3 4- i*r)} 

corresponds to a primary wave of type 

6 cos 2 a 

* " 6 8 T(5Ty) a ~"fT ' 

[H. Lamb.] 

4. Verify that with the function </> in Example 3 

/QO 
<f> dt = TT/C, 
-oo 

at all points in the field. rTT _ 

[H. Lamb.] 

11-7. For a discussion of other diffraction problems reference may be 
made to G. Wolfsohn's article, "Strenge Theorie der Interferenz und 
Beugung," Handbuch der Physik, Bd. xx, T. 7. In this article accounts 
are given of the work of Schaefer and others on the diffraction of un- 
damped electric waves by a dielectric circular cylinder and of Schwarz- 
schild's treatment of diffraction by a slit [Math. Ann. Bd. LV, S. 177 (1902)]. 
A study of diffraction by an elliptic cylinder has been commenced by 
Sieger*. For this study and for an analogous study of diffraction by a 
hyperbolic cylinder the substitution x -f iy = h cosh ( + irj) may be used 
with advantage and then for the representation of divergent waves it is 
necessary to find solutions of Mathieu's equation which v